Represents a 3-dimensional triangulation, typically of a 3-manifold. More...
#include <triangulation/dim3.h>
Public Types | |
| using | TuraevViroSet = std::map< std::pair< unsigned long, bool >, Cyclotomic > |
| A map from (r, parity) pairs to Turaev-Viro invariants, as described by turaevViro(). More... | |
Public Member Functions | |
| std::shared_ptr< PacketOf< Triangulation< dim > > > | packet () |
| Returns the packet that holds this data, if there is one. More... | |
| std::shared_ptr< const PacketOf< Triangulation< dim > > > | packet () const |
| Returns the packet that holds this data, if there is one. More... | |
| std::string | anonID () const |
| A unique string ID that can be used in place of a packet ID. More... | |
| std::string | str () const |
| Returns a short text representation of this object. More... | |
| std::string | utf8 () const |
| Returns a short text representation of this object using unicode characters. More... | |
| std::string | detail () const |
| Returns a detailed text representation of this object. More... | |
Constructors and Destructors | |
| Triangulation ()=default | |
| Default constructor. More... | |
| Triangulation (const Triangulation< 3 > ©) | |
| Creates a new copy of the given triangulation. More... | |
| Triangulation (const Triangulation ©, bool cloneProps) | |
| Creates a new copy of the given triangulation, with the option of whether or not to clone its computed properties also. More... | |
| Triangulation (Triangulation &&src) noexcept=default | |
| Moves the given triangulation into this new triangulation. More... | |
| Triangulation (const Link &link, bool simplify=true) | |
| Creates a new ideal triangulation representing the complement of the given link in the 3-sphere. More... | |
| Triangulation (const std::string &description) | |
| "Magic" constructor that tries to find some way to interpret the given string as a triangulation. More... | |
| Triangulation (snappy.Manifold m) | |
| Python-only constructor that copies the given SnapPy manifold. More... | |
| Triangulation (snappy.Triangulation t) | |
| Python-only constructor that copies the given SnapPy triangulation. More... | |
| ~Triangulation () | |
| Destroys this triangulation. More... | |
| std::shared_ptr< Packet > | inAnyPacket () |
| Returns the packet that holds this data, even if it is held indirectly via a SnapPea triangulation. More... | |
| std::shared_ptr< const Packet > | inAnyPacket () const |
| Returns the packet that holds this data, even if it is held indirectly via a SnapPea triangulation. More... | |
| SnapPeaTriangulation * | isSnapPea () |
| Returns the SnapPea triangulation that holds this data, if there is one. More... | |
| const SnapPeaTriangulation * | isSnapPea () const |
| Returns the SnapPea triangulation that holds this data, if there is one. More... | |
Tetrahedra | |
| Tetrahedron< 3 > * | newTetrahedron () |
| A dimension-specific alias for newSimplex(). More... | |
| Tetrahedron< 3 > * | newTetrahedron (const std::string &desc) |
| A dimension-specific alias for newSimplex(). More... | |
| template<int k> | |
| std::array< Tetrahedron< 3 > *, k > | newTetrahedra () |
| A dimension-specific alias for newSimplices(). More... | |
| void | newTetrahedra (size_t k) |
| A dimension-specific alias for newSimplices(). More... | |
| void | removeTetrahedron (Tetrahedron< 3 > *tet) |
| A dimension-specific alias for removeSimplex(). More... | |
| void | removeTetrahedronAt (size_t index) |
| A dimension-specific alias for removeSimplexAt(). More... | |
| void | removeAllTetrahedra () |
| A dimension-specific alias for removeAllSimplices(). More... | |
| Triangulation & | operator= (const Triangulation &src) |
| Sets this to be a (deep) copy of the given triangulation. More... | |
| Triangulation & | operator= (Triangulation &&src) |
| Moves the contents of the given triangulation into this triangulation. More... | |
| void | swap (Triangulation< 3 > &other) |
| Swaps the contents of this and the given triangulation. More... | |
| void | swapContents (Triangulation< 3 > &other) |
| Deprecated routine that swaps the contents of this and the given triangulation. More... | |
Skeletal Queries | |
| bool | hasBoundaryTriangles () const |
| A dimension-specific alias for hasBoundaryFacets(). More... | |
| size_t | countBoundaryTriangles () const |
| A dimension-specific alias for countBoundaryFacets(). More... | |
| bool | hasTwoSphereBoundaryComponents () const |
| Determines if this triangulation contains any two-sphere boundary components. More... | |
| bool | hasNegativeIdealBoundaryComponents () const |
| Determines if this triangulation contains any ideal boundary components with negative Euler characteristic. More... | |
Basic Properties | |
| long | eulerCharManifold () const |
| Returns the Euler characteristic of the corresponding compact 3-manifold. More... | |
| bool | isIdeal () const |
| Determines if this triangulation is ideal. More... | |
| bool | isStandard () const |
| Determines if this triangulation is standard. More... | |
| bool | isClosed () const |
| Determines if this triangulation is closed. More... | |
| bool | isOrdered () const |
| Determines if this triangulation is ordered; that is, if tetrahedron vertices are labelled so that all gluing permutations are order-preserving on the tetrahedron faces. More... | |
Algebraic Properties | |
| const AbelianGroup & | homologyRel () const |
| Returns the relative first homology group with respect to the boundary for this triangulation. More... | |
| const AbelianGroup & | homologyBdry () const |
| Returns the first homology group of the boundary for this triangulation. More... | |
| AbelianGroup | homologyH2 () const |
| Deprecated routine that returns the second homology group for this triangulation. More... | |
| unsigned long | homologyH2Z2 () const |
| Returns the second homology group with coefficients in Z_2 for this triangulation. More... | |
| Cyclotomic | turaevViro (unsigned long r, bool parity=true, Algorithm alg=ALG_DEFAULT, ProgressTracker *tracker=nullptr) const |
| Computes the given Turaev-Viro state sum invariant of this 3-manifold using exact arithmetic. More... | |
| double | turaevViroApprox (unsigned long r, unsigned long whichRoot=1, Algorithm alg=ALG_DEFAULT) const |
| Computes the given Turaev-Viro state sum invariant of this 3-manifold using a fast but inexact floating-point approximation. More... | |
| const TuraevViroSet & | allCalculatedTuraevViro () const |
| Returns the cache of all Turaev-Viro state sum invariants that have been calculated for this 3-manifold. More... | |
| Edge< 3 > * | longitude () |
| Modifies a triangulated knot complement so that the algebraic longitude follows a single boundary edge, and returns this edge. More... | |
| std::pair< Edge< 3 > *, Edge< 3 > * > | meridianLongitude () |
| Modifies a triangulated knot complement so that the meridian and algebraic longitude each follow a single boundary edge, and returns these two edges. More... | |
Normal Surfaces and Angle Structures | |
| bool | isZeroEfficient () const |
| Determines if this triangulation is 0-efficient. More... | |
| bool | knowsZeroEfficient () const |
| Is it already known whether or not this triangulation is 0-efficient? See isZeroEfficient() for further details. More... | |
| bool | hasSplittingSurface () const |
| Determines whether this triangulation has a normal splitting surface. More... | |
| std::optional< NormalSurface > | nonTrivialSphereOrDisc () const |
| Searches for a non-vertex-linking normal sphere or disc within this triangulation. More... | |
| std::optional< NormalSurface > | hasNonTrivialSphereOrDisc () const |
| A deprecated alias for nonTrivialSphereOrDisc(), which searches for a non-vertex-linking normal sphere or disc within this triangulation. More... | |
| std::optional< NormalSurface > | octagonalAlmostNormalSphere () const |
| Searches for an octagonal almost normal 2-sphere within this triangulation. More... | |
| std::optional< NormalSurface > | hasOctagonalAlmostNormalSphere () const |
| A deprecated alias for octagonalAlmostNormalSphere(), which searches for an octagonal almost normal 2-sphere within this triangulation. More... | |
| const AngleStructure & | strictAngleStructure () const |
| Returns a strict angle structure on this triangulation, if one exists. More... | |
| const AngleStructure & | findStrictAngleStructure () const |
| A deprecated alias for strictAngleStructure(), which returns a strict angle structure on this triangulation if one exists. More... | |
| bool | hasStrictAngleStructure () const |
| Determines whether this triangulation supports a strict angle structure. More... | |
| bool | knowsStrictAngleStructure () const |
| Is it already known (or trivial to determine) whether or not this triangulation supports a strict angle structure? See hasStrictAngleStructure() for further details. More... | |
| const AngleStructure & | generalAngleStructure () const |
| Returns a generalised angle structure on this triangulation, if one exists. More... | |
| bool | hasGeneralAngleStructure () const |
| Determines whether this triangulation supports a generalised angle structure. More... | |
Skeletal Transformations | |
| std::set< Edge< 3 > * > | maximalForestInBoundary () const |
| Produces a maximal forest in the 1-skeleton of the triangulation boundary. More... | |
| std::set< Edge< 3 > * > | maximalForestInSkeleton (bool canJoinBoundaries=true) const |
| Produces a maximal forest in the triangulation's 1-skeleton. More... | |
| bool | intelligentSimplify () |
| Attempts to simplify the triangulation using fast and greedy heuristics. More... | |
| bool | simplifyToLocalMinimum (bool perform=true) |
| Uses all known simplification moves to reduce the triangulation monotonically to some local minimum number of tetrahedra. More... | |
| bool | simplifyExhaustive (int height=1, unsigned nThreads=1, ProgressTrackerOpen *tracker=nullptr) |
| Attempts to simplify this triangulation using a slow but exhaustive search through the Pachner graph. More... | |
| template<typename Action , typename... Args> | |
| bool | retriangulate (int height, unsigned nThreads, ProgressTrackerOpen *tracker, Action &&action, Args &&... args) const |
| Explores all triangulations that can be reached from this via Pachner moves, without exceeding a given number of additional tetrahedra. More... | |
| bool | minimiseBoundary () |
| Ensures that the boundary contains the smallest possible number of triangles, potentially adding tetrahedra to do this. More... | |
| bool | minimizeBoundary () |
| A synonym for minimiseBoundary(). More... | |
| bool | fourFourMove (Edge< 3 > *e, int newAxis, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a 4-4 move about the given edge. More... | |
| bool | twoZeroMove (Edge< 3 > *e, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a 2-0 move about the given edge of degree 2. More... | |
| bool | twoZeroMove (Vertex< 3 > *v, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a 2-0 move about the given vertex of degree 2. More... | |
| bool | twoOneMove (Edge< 3 > *e, int edgeEnd, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a 2-1 move about the given edge. More... | |
| bool | zeroTwoMove (EdgeEmbedding< 3 > e0, int t0, EdgeEmbedding< 3 > e1, int t1, bool check=true, bool perform=true) |
Checks the eligibility of and/or performs a 0-2 move about the (not necessarily distinct) triangles e0.tetrahedron()->triangle( e0.vertices()[t0] ) and e1.tetrahedron()->triangle( e1.vertices()[t1] ). More... | |
| bool | zeroTwoMove (Edge< 3 > *e, size_t t0, size_t t1, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a 0-2 move about the (not necessarily distinct) triangles incident to e that are numbered t0 and t1. More... | |
| bool | zeroTwoMove (Triangle< 3 > *t0, int e0, Triangle< 3 > *t1, int e1, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a 0-2 move about the (not necessarily distinct) triangles t0 and t1. More... | |
| bool | openBook (Triangle< 3 > *t, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a book opening move about the given triangle. More... | |
| bool | closeBook (Edge< 3 > *e, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a book closing move about the given boundary edge. More... | |
| bool | shellBoundary (Tetrahedron< 3 > *t, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a boundary shelling move on the given tetrahedron. More... | |
| bool | collapseEdge (Edge< 3 > *e, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a collapse of an edge between two distinct vertices. More... | |
| void | reorderTetrahedraBFS (bool reverse=false) |
| Reorders the tetrahedra of this triangulation using a breadth-first search, so that small-numbered tetrahedra are adjacent to other small-numbered tetrahedra. More... | |
| bool | order (bool forceOriented=false) |
| Relabels tetrahedron vertices in this triangulation to give an ordered triangulation, if possible. More... | |
Decompositions | |
| std::vector< Triangulation< 3 > > | summands () const |
| Computes the connected sum decomposition of this triangulation. More... | |
| bool | isSphere () const |
| Determines whether this is a triangulation of a 3-sphere. More... | |
| bool | isThreeSphere () const |
| Deprecated function to test if this is a 3-sphere triangulation. More... | |
| bool | knowsSphere () const |
| Is it already known (or trivial to determine) whether or not this is a triangulation of a 3-sphere? See isSphere() for further details. More... | |
| bool | knowsThreeSphere () const |
| Deprecated function to determine if it is already known (or trivial to determine) whether this is a 3-sphere triangulation. More... | |
| bool | isBall () const |
| Determines whether this is a triangulation of a 3-dimensional ball. More... | |
| bool | knowsBall () const |
| Is it already known (or trivial to determine) whether or not this is a triangulation of a 3-dimensional ball? See isBall() for further details. More... | |
| bool | isSolidTorus () const |
| Determines whether this is a triangulation of the solid torus; that is, the unknot complement. More... | |
| bool | knowsSolidTorus () const |
| Is it already known (or trivial to determine) whether or not this is a triangulation of a solid torus (that is, the unknot complement)? See isSolidTorus() for further details. More... | |
| bool | isTxI () const |
| Determines whether or not the underlying 3-manifold is the product of a torus with an interval. More... | |
| bool | knowsTxI () const |
| Is it already known (or trivial to determine) whether or not this is a triangulation of a the product of a torus with an interval? See isTxI() for further details. More... | |
| bool | isIrreducible () const |
| Determines whether the underlying 3-manifold (which must be closed) is irreducible. More... | |
| bool | knowsIrreducible () const |
| Is it already known (or trivial to determine) whether or not the underlying 3-manifold is irreducible? See isIrreducible() for further details. More... | |
| bool | hasCompressingDisc () const |
| Searches for a compressing disc within the underlying 3-manifold. More... | |
| bool | knowsCompressingDisc () const |
| Is it already known (or trivial to determine) whether or not the underlying 3-manifold contains a compressing disc? See hasCompressingDisc() for further details. More... | |
| bool | isHaken () const |
| Determines whether the underlying 3-manifold (which must be closed and orientable) is Haken. More... | |
| bool | knowsHaken () const |
| Is it already known (or trivial to determine) whether or not the underlying 3-manifold is Haken? See isHaken() for further details. More... | |
| bool | hasSimpleCompressingDisc () const |
| Searches for a "simple" compressing disc inside this triangulation. More... | |
| const TreeDecomposition & | niceTreeDecomposition () const |
| Returns a nice tree decomposition of the face pairing graph of this triangulation. More... | |
Subdivisions, Extensions and Covers | |
| bool | idealToFinite () |
| Converts an ideal triangulation into a finite triangulation. More... | |
| void | pinchEdge (Edge< 3 > *e) |
| Pinches an internal edge to a point. More... | |
| void | puncture (Tetrahedron< 3 > *tet=nullptr) |
| Punctures this manifold by removing a 3-ball from the interior of the given tetrahedron. More... | |
Building Triangulations | |
| Tetrahedron< 3 > * | layerOn (Edge< 3 > *edge) |
| Performs a layering upon the given boundary edge of the triangulation. More... | |
| bool | fillTorus (unsigned long cuts0, unsigned long cuts1, unsigned long cuts2, BoundaryComponent< 3 > *bc=nullptr) |
| Fills a two-triangle torus boundary component by attaching a solid torus along a given curve. More... | |
| bool | fillTorus (Edge< 3 > *e0, Edge< 3 > *e1, Edge< 3 > *e2, unsigned long cuts0, unsigned long cuts1, unsigned long cuts2) |
| Fills a two-triangle torus boundary component by attaching a solid torus along a given curve. More... | |
| Tetrahedron< 3 > * | insertLayeredSolidTorus (unsigned long cuts0, unsigned long cuts1) |
| Inserts a new layered solid torus into the triangulation. More... | |
| void | insertLayeredLensSpace (size_t p, size_t q) |
| Deprecated routine that inserts a new layered lens space L(p,q) into this triangulation. More... | |
| void | insertLayeredLoop (size_t length, bool twisted) |
| Deprecated routine that inserts a layered loop of the given length into this triangulation. More... | |
| void | insertAugTriSolidTorus (long a1, long b1, long a2, long b2, long a3, long b3) |
| Deprecated routine that inserts an augmented triangular solid torus with the given parameters into this triangulation. More... | |
| void | insertSFSOverSphere (long a1=1, long b1=0, long a2=1, long b2=0, long a3=1, long b3=0) |
| Deprecated routine that inserts an orientable Seifert fibred space with at most three exceptional fibres over the 2-sphere into this triangulation. More... | |
| void | connectedSumWith (const Triangulation &other) |
| Forms the connected sum of this triangulation with the given triangulation. More... | |
| void | insertRehydration (const std::string &dehydration) |
| Deprecated routine that inserts the rehydration of the given string into this triangulation. More... | |
Exporting Triangulations | |
| std::string | dehydrate () const |
| Dehydrates this triangulation into an alphabetical string. More... | |
| std::string | snapPea () const |
| Returns a string containing the full contents of a SnapPea data file that describes this triangulation. More... | |
| void | snapPea (std::ostream &out) const |
| Writes the full contents of a SnapPea data file describing this triangulation to the given output stream. More... | |
| bool | saveSnapPea (const char *filename) const |
| Writes this triangulation to the given file using SnapPea's native file format. More... | |
| std::string | recogniser () const |
| Returns a string that expresses this triangulation in Matveev's 3-manifold recogniser format. More... | |
| std::string | recognizer () const |
| A synonym for recogniser(). More... | |
| void | recogniser (std::ostream &out) const |
| Writes a string expressing this triangulation in Matveev's 3-manifold recogniser format to the given output stream. More... | |
| void | recognizer (std::ostream &out) const |
| A synonym for recognizer(std::ostream&). More... | |
| bool | saveRecogniser (const char *filename) const |
| Writes this triangulation to the given file in Matveev's 3-manifold recogniser format. More... | |
| bool | saveRecognizer (const char *filename) const |
| A synonym for saveRecogniser(). More... | |
Simplices | |
| size_t | size () const |
| Returns the number of top-dimensional simplices in the triangulation. More... | |
| auto | simplices () const |
| Returns an object that allows iteration through and random access to all top-dimensional simplices in this triangulation. More... | |
| Simplex< dim > * | simplex (size_t index) |
| Returns the top-dimensional simplex at the given index in the triangulation. More... | |
| const Simplex< dim > * | simplex (size_t index) const |
| Returns the top-dimensional simplex at the given index in the triangulation. More... | |
| Simplex< dim > * | newSimplex () |
| Creates a new top-dimensional simplex and adds it to this triangulation. More... | |
| Simplex< dim > * | newSimplex (const std::string &desc) |
| Creates a new top-dimensional simplex with the given description and adds it to this triangulation. More... | |
| std::array< Simplex< dim > *, k > | newSimplices () |
| Creates k new top-dimensional simplices, adds them to this triangulation, and returns them in a std::array. More... | |
| void | newSimplices (size_t k) |
| Creates k new top-dimensional simplices and adds them to this triangulation. More... | |
| void | removeSimplex (Simplex< dim > *simplex) |
| Removes the given top-dimensional simplex from this triangulation. More... | |
| void | removeSimplexAt (size_t index) |
| Removes the top-dimensional simplex at the given index in this triangulation. More... | |
| void | removeAllSimplices () |
| Removes all simplices from the triangulation. More... | |
| void | moveContentsTo (Triangulation< dim > &dest) |
| Moves the contents of this triangulation into the given destination triangulation, without destroying any pre-existing contents. More... | |
Skeletal Queries | |
| size_t | countComponents () const |
| Returns the number of connected components in this triangulation. More... | |
| size_t | countBoundaryComponents () const |
| Returns the number of boundary components in this triangulation. More... | |
| size_t | countFaces () const |
| Returns the number of subdim-faces in this triangulation. More... | |
| size_t | countFaces (int subdim) const |
| Returns the number of subdim-faces in this triangulation, where the face dimension does not need to be known until runtime. More... | |
| size_t | countVertices () const |
| A dimension-specific alias for countFaces<0>(). More... | |
| size_t | countEdges () const |
| A dimension-specific alias for countFaces<1>(). More... | |
| size_t | countTriangles () const |
| A dimension-specific alias for countFaces<2>(). More... | |
| size_t | countTetrahedra () const |
| A dimension-specific alias for countFaces<3>(). More... | |
| size_t | countPentachora () const |
| A dimension-specific alias for countFaces<4>(). More... | |
| std::vector< size_t > | fVector () const |
| Returns the f-vector of this triangulation, which counts the number of faces of all dimensions. More... | |
| auto | components () const |
| Returns an object that allows iteration through and random access to all components of this triangulation. More... | |
| auto | boundaryComponents () const |
| Returns an object that allows iteration through and random access to all boundary components of this triangulation. More... | |
| auto | faces () const |
| Returns an object that allows iteration through and random access to all subdim-faces of this triangulation, in a way that is optimised for C++ programmers. More... | |
| auto | faces (int subdim) const |
| Returns an object that allows iteration through and random access to all subdim-faces of this triangulation, in a way that is optimised for Python programmers. More... | |
| auto | vertices () const |
| A dimension-specific alias for faces<0>(). More... | |
| auto | edges () const |
| A dimension-specific alias for faces<1>(). More... | |
| auto | triangles () const |
| A dimension-specific alias for faces<2>(), or an alias for simplices() in dimension dim = 2. More... | |
| auto | tetrahedra () const |
| A dimension-specific alias for faces<3>(), or an alias for simplices() in dimension dim = 3. More... | |
| auto | pentachora () const |
| A dimension-specific alias for faces<4>(), or an alias for simplices() in dimension dim = 4. More... | |
| Component< dim > * | component (size_t index) const |
| Returns the requested connected component of this triangulation. More... | |
| BoundaryComponent< dim > * | boundaryComponent (size_t index) const |
| Returns the requested boundary component of this triangulation. More... | |
| Face< dim, subdim > * | face (size_t index) const |
| Returns the requested subdim-face of this triangulation, in a way that is optimised for C++ programmers. More... | |
| auto | face (int subdim, size_t index) const |
| Returns the requested subdim-face of this triangulation, in a way that is optimised for Python programmers. More... | |
| Face< dim, 0 > * | vertex (size_t index) const |
| A dimension-specific alias for face<0>(). More... | |
| Face< dim, 1 > * | edge (size_t index) const |
| A dimension-specific alias for face<1>(). More... | |
| Face< dim, 2 > * | triangle (size_t index) |
| A dimension-specific alias for face<2>(), or an alias for simplex() in dimension dim = 2. More... | |
| auto | triangle (size_t index) const |
| A dimension-specific alias for face<2>(), or an alias for simplex() in dimension dim = 2. More... | |
| Face< dim, 3 > * | tetrahedron (size_t index) |
| A dimension-specific alias for face<3>(), or an alias for simplex() in dimension dim = 3. More... | |
| auto | tetrahedron (size_t index) const |
| A dimension-specific alias for face<3>(), or an alias for simplex() in dimension dim = 3. More... | |
| Face< dim, 4 > * | pentachoron (size_t index) |
| A dimension-specific alias for face<4>(), or an alias for simplex() in dimension dim = 4. More... | |
| auto | pentachoron (size_t index) const |
| A dimension-specific alias for face<4>(), or an alias for simplex() in dimension dim = 4. More... | |
Basic Properties | |
| bool | isEmpty () const |
| Determines whether this triangulation is empty. More... | |
| bool | isValid () const |
| Determines if this triangulation is valid. More... | |
| bool | hasBoundaryFacets () const |
| Determines if this triangulation has any boundary facets. More... | |
| size_t | countBoundaryFacets () const |
| Returns the total number of boundary facets in this triangulation. More... | |
| bool | isOrientable () const |
| Determines if this triangulation is orientable. More... | |
| bool | isConnected () const |
| Determines if this triangulation is connected. More... | |
| bool | isOriented () const |
| Determines if this triangulation is oriented; that is, if the vertices of its top-dimensional simplices are labelled in a way that preserves orientation across adjacent facets. More... | |
| long | eulerCharTri () const |
| Returns the Euler characteristic of this triangulation. More... | |
Algebraic Properties | |
| const GroupPresentation & | fundamentalGroup () const |
| Returns the fundamental group of this triangulation. More... | |
| void | simplifiedFundamentalGroup (GroupPresentation newGroup) |
| Notifies the triangulation that you have simplified the presentation of its fundamental group. More... | |
| AbelianGroup | homology () const |
| Returns the kth homology group of this triangulation, treating any ideal vertices as though they had been truncated. More... | |
| AbelianGroup | homology (int k) const |
| Returns the kth homology group of this triangulation, treating any ideal vertices as though they had been truncated, where the parameter k does not need to be known until runtime. More... | |
| const AbelianGroup & | homologyH1 () const |
| Deprecated routine that returns the first homology group for this triangulation. More... | |
| MarkedAbelianGroup | markedHomology () const |
| Returns the kth homology group of this triangulation, without truncating ideal vertices, but with explicit coordinates that track the individual k-faces of this triangulation. More... | |
| MarkedAbelianGroup | markedHomology (int k) const |
| Returns the kth homology group of this triangulation, without truncating ideal vertices, but with explicit coordinates that track the individual k-faces of this triangulation, where the parameter k does not need to be known until runtime. More... | |
| MatrixInt | boundaryMap () const |
| Returns the boundary map from subdim-faces to (subdim-1)-faces of the triangulation. More... | |
| MatrixInt | boundaryMap (int subdim) const |
| Returns the boundary map from subdim-faces to (subdim-1)-faces of the triangulation, where the face dimension does not need to be known until runtime. More... | |
Skeletal Transformations | |
| void | orient () |
| Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices are oriented consistently, if possible. More... | |
| void | reflect () |
| Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices reflect their orientation. More... | |
| bool | pachner (Face< dim, k > *f, bool check=true, bool perform=true) |
| Checks the eligibility of and/or performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face. More... | |
Subdivisions, Extensions and Covers | |
| void | makeDoubleCover () |
| Converts this triangulation into its double cover. More... | |
| void | barycentricSubdivision () |
| Does a barycentric subdivision of the triangulation. More... | |
| bool | finiteToIdeal () |
| Converts each real boundary component into a cusp (i.e., an ideal vertex). More... | |
Decompositions | |
| std::vector< Triangulation< dim > > | triangulateComponents () const |
| Returns the individual connected components of this triangulation. More... | |
Isomorphism Testing | |
| bool | operator== (const Triangulation< dim > &other) const |
| Determines if this triangulation is combinatorially identical to the given triangulation. More... | |
| bool | operator!= (const Triangulation< dim > &other) const |
| Determines if this triangulation is not combinatorially identical to the given triangulation. More... | |
| bool | isIdenticalTo (const Triangulation< dim > &other) const |
| Deprecated routine that determines if this triangulation is combinatorially identical to the given triangulation. More... | |
| std::optional< Isomorphism< dim > > | isIsomorphicTo (const Triangulation< dim > &other) const |
| Determines if this triangulation is combinatorially isomorphic to the given triangulation. More... | |
| std::optional< Isomorphism< dim > > | isContainedIn (const Triangulation< dim > &other) const |
| Determines if an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components). More... | |
| bool | findAllIsomorphisms (const Triangulation< dim > &other, Action &&action, Args &&... args) const |
| Finds all ways in which this triangulation is combinatorially isomorphic to the given triangulation. More... | |
| bool | findAllSubcomplexesIn (const Triangulation< dim > &other, Action &&action, Args &&... args) const |
| Finds all ways in which an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components). More... | |
| bool | makeCanonical () |
| Relabel the top-dimensional simplices and their vertices so that this triangulation is in canonical form. More... | |
Building Triangulations | |
| void | insertTriangulation (const Triangulation< dim > &source) |
| Inserts a copy of the given triangulation into this triangulation. More... | |
| void | insertConstruction (size_t nSimplices, const int adjacencies[][dim+1], const int gluings[][dim+1][dim+1]) |
| Deprecated routine that inserts a given triangulation into this triangulation, where the given triangulation is described by a pair of integer arrays. More... | |
Exporting Triangulations | |
| void | writeTextShort (std::ostream &out) const |
| Writes a short text representation of this object to the given output stream. More... | |
| void | writeTextLong (std::ostream &out) const |
| Writes a detailed text representation of this object to the given output stream. More... | |
| Encoding::Signature | isoSig () const |
| Constructs the isomorphism signature of the given type for this triangulation. More... | |
| std::pair< typename Encoding::Signature, Isomorphism< dim > > | isoSigDetail () const |
| Constructs the isomorphism signature for this triangulation, along with the relabelling that will occur when the triangulation is reconstructed from it. More... | |
| std::string | dumpConstruction () const |
| Returns C++ code that can be used to reconstruct this triangulation. More... | |
Static Public Attributes | |
| static constexpr int | dimension |
| A compile-time constant that gives the dimension of the triangulation. More... | |
Protected Member Functions | |
| void | swap (Snapshottable &other) noexcept |
| Swap operation. More... | |
| void | takeSnapshot () |
| Must be called before modification and/or destruction of the type T contents. More... | |
| bool | isReadOnlySnapshot () const |
| Determines if this object is a read-only deep copy that was created by a snapshot. More... | |
Protected Attributes | |
| MarkedVector< Simplex< dim > > | simplices_ |
| The top-dimensional simplices that form the triangulation. More... | |
| MarkedVector< BoundaryComponent< dim > > | boundaryComponents_ |
| The components that form the boundary of the triangulation. More... | |
| bool | valid_ |
| Is this triangulation valid? See isValid() for details on what this means. More... | |
| uint8_t | topologyLock_ |
| If non-zero, this will cause Triangulation<dim>::clearAllProperties() to preserve any computed properties that related to the manifold (as opposed to the specific triangulation). More... | |
| PacketHeldBy | heldBy_ |
| Indicates whether this Held object is in fact the inherited data for a PacketOf<Held>. More... | |
Importing Triangulations | |
| class | regina::Face< 3, 3 > |
| class | regina::detail::SimplexBase< 3 > |
| class | regina::detail::TriangulationBase< 3 > |
| class | PacketData< Triangulation< 3 > > |
| class | regina::XMLTriangulationReader< 3 > |
| class | regina::XMLWriter< Triangulation< 3 > > |
| static Triangulation< 3 > | enterTextTriangulation (std::istream &in, std::ostream &out) |
| Deprecated function that allows the user to interactively enter a triangulation in plain text. More... | |
| static Triangulation< 3 > | rehydrate (const std::string &dehydration) |
| Rehydrates the given alphabetical string into a 3-dimensional triangulation. More... | |
| static Triangulation< 3 > | fromSnapPea (const std::string &snapPeaData) |
| Extracts the tetrahedron gluings from a string that contains the full contents of a SnapPea data file. More... | |
Importing Triangulations | |
| static Triangulation< dim > | fromGluings (size_t size, std::initializer_list< std::tuple< size_t, int, size_t, Perm< dim+1 > > > gluings) |
| Creates a triangulation from a hard-coded list of gluings. More... | |
| static Triangulation< dim > | fromGluings (size_t size, Iterator beginGluings, Iterator endGluings) |
| Creates a triangulation from a list of gluings. More... | |
| static Triangulation< dim > | fromIsoSig (const std::string &sig) |
| Recovers a full triangulation from an isomorphism signature. More... | |
| static Triangulation< dim > | fromSig (const std::string &sig) |
| Alias for fromIsoSig(), to recover a full triangulation from an isomorphism signature. More... | |
| static size_t | isoSigComponentSize (const std::string &sig) |
| Deduces the number of top-dimensional simplices in a connected triangulation from its isomorphism signature. More... | |
| void | ensureSkeleton () const |
| Ensures that all "on demand" skeletal objects have been calculated. More... | |
| bool | calculatedSkeleton () const |
| Determines whether the skeletal objects and properties of this triangulation have been calculated. More... | |
| void | clearBaseProperties () |
| Clears all properties that are managed by this base class. More... | |
| void | swapBaseData (TriangulationBase< dim > &other) |
| Swaps all data that is managed by this base class, including simplices, skeletal data, cached properties and the snapshotting data, with the given triangulation. More... | |
| void | writeXMLBaseProperties (std::ostream &out) const |
| Writes a chunk of XML containing properties of this triangulation. More... | |
Detailed Description
Represents a 3-dimensional triangulation, typically of a 3-manifold.
This is a specialisation of the generic Triangulation class template; see the Triangulation documentation for a general overview of how the triangulation classes work.
This 3-dimensional specialisation offers significant extra functionality, including many functions specific to 3-manifolds.
This class implements C++ move semantics and adheres to the C++ Swappable requirement. It is designed to avoid deep copies wherever possible, even when passing or returning objects by value.
- Todo:
Feature: Is the boundary incompressible?
Feature (long-term): Am I obviously a handlebody? (Simplify and see if there is nothing left). Am I obviously not a handlebody? (Compare homology with boundary homology).
Feature (long-term): Is the triangulation Haken?
Feature (long-term): What is the Heegaard genus?
Feature (long-term): Have a subcomplex as a new type. Include routines to crush a subcomplex or to expand a subcomplex to a normal surface.
Member Typedef Documentation
◆ TuraevViroSet
| using regina::Triangulation< 3 >::TuraevViroSet = std::map<std::pair<unsigned long, bool>, Cyclotomic> |
A map from (r, parity) pairs to Turaev-Viro invariants, as described by turaevViro().
Constructor & Destructor Documentation
◆ Triangulation() [1/8]
|
default |
Default constructor.
Creates an empty triangulation.
◆ Triangulation() [2/8]
|
inline |
Creates a new copy of the given triangulation.
This will clone any computed properties (such as homology, fundamental group, and so on) of the given triangulation also. If you want a "clean" copy that resets all properties to unknown, you can use the two-argument copy constructor instead.
- Parameters
-
copy the triangulation to copy.
◆ Triangulation() [3/8]
| regina::Triangulation< 3 >::Triangulation | ( | const Triangulation< 3 > & | copy, |
| bool | cloneProps | ||
| ) |
Creates a new copy of the given triangulation, with the option of whether or not to clone its computed properties also.
- Parameters
-
copy the triangulation to copy. cloneProps trueif this should also clone any computed properties of the given triangulation (such as homology, fundamental group, and so on), orfalseif the new triangulation should have all properties marked as unknown.
◆ Triangulation() [4/8]
|
defaultnoexcept |
Moves the given triangulation into this new triangulation.
This is much faster than the copy constructor, but is still linear time. This is because every tetrahedron must be adjusted to point back to this new triangulation instead of src.
All tetrahedra and skeletal objects (faces, components and boundary components) that belong to src will be moved into this triangulation, and so any pointers or references to Tetrahedron<3>, Face<3, subdim>, Component<3> or BoundaryComponent<3> objects will remain valid. Likewise, all cached properties will be moved into this triangulation.
The triangulation that is passed (src) will no longer be usable.
- Note
- This operator is marked
noexcept, and in particular does not fire any change events. This is because this triangulation is freshly constructed (and therefore has no listeners yet), and because we assume that src is about to be destroyed (an action that will fire a packet destruction event).
- Parameters
-
src the triangulation to move.
◆ Triangulation() [5/8]
| regina::Triangulation< 3 >::Triangulation | ( | const Link & | link, |
| bool | simplify = true |
||
| ) |
Creates a new ideal triangulation representing the complement of the given link in the 3-sphere.
The triangulation will have one ideal vertex for each link component. Assuming you pass simplify as true (the default), there will typically be no internal vertices; however, this is not guaranteed.
Initially, each tetrahedron will be oriented according to a right-hand rule: the thumb of the right hand points from vertices 0 to 1, and the fingers curl around to point from vertices 2 to 3. If you pass simplify as true, then Regina will attempt to simplify the triangulation to as few tetrahedra as possible: this may relabel the tetrahedra, though their orientations will be preserved.
This is the same triangulation that is produced by Link::complement().
- Parameters
-
link the link whose complement we should build. simplify trueif and only if the triangulation should be simplified to use as few tetrahedra as possible.
◆ Triangulation() [6/8]
| regina::Triangulation< 3 >::Triangulation | ( | const std::string & | description | ) |
"Magic" constructor that tries to find some way to interpret the given string as a triangulation.
At present, Regina understands the following types of strings (and attempts to parse them in the following order):
- isomorphism signatures (see fromIsoSig());
- dehydration strings (see rehydrate());
- the contents of a SnapPea data file (see fromSnapPea()).
This list may grow in future versions of Regina.
- Warning
- If you pass the contents of a SnapPea data file, then only the tetrahedron gluings will be read; all other SnapPea-specific information (such as peripheral curves) will be lost. See fromSnapPea() for details, and for other alternatives that preserve SnapPea-specific data.
- Exceptions
-
InvalidArgument Regina could not interpret the given string as representing a triangulation using any of the supported string types.
- Parameters
-
description a string that describes a 3-manifold triangulation.
◆ Triangulation() [7/8]
| regina::Triangulation< 3 >::Triangulation | ( | snappy.Manifold | m | ) |
Python-only constructor that copies the given SnapPy manifold.
- Warning
- Only the tetrahedron gluings will be copied; all other SnapPy-specific information (such as peripheral curves) will be lost. See fromSnapPea() for details, and for other alternatives that preserve SnapPy-specific data.
- C++
- Not present.
- Parameters
-
m a SnapPy object of type snappy.Manifold.
◆ Triangulation() [8/8]
| regina::Triangulation< 3 >::Triangulation | ( | snappy.Triangulation< 3 > | t | ) |
Python-only constructor that copies the given SnapPy triangulation.
- Warning
- Only the tetrahedron gluings will be copied; all other SnapPy-specific information (such as peripheral curves) will be lost. See fromSnapPea() for details, and for other alternatives that preserve SnapPy-specific data.
- C++
- Not present.
- Parameters
-
t a SnapPy object of type snappy.Triangulation.
◆ ~Triangulation()
|
inline |
Destroys this triangulation.
The constituent tetrahedra, the cellular structure and all other properties will also be destroyed.
Member Function Documentation
◆ allCalculatedTuraevViro()
|
inline |
Returns the cache of all Turaev-Viro state sum invariants that have been calculated for this 3-manifold.
This cache is updated every time turaevViro() is called, and is emptied whenever the triangulation is modified.
Turaev-Viro invariants are identified by an (r, parity) pair as described in the turaevViro() documentation. The cache is just a set that maps (r, parity) pairs to the corresponding invariant values.
For even values of r, the parity is ignored when calling turaevViro() (since the even and odd versions of the invariant contain essentially the same information). Therefore, in this cache, all even values of r will have the corresponding parities set to false.
- Note
- All invariants in this cache are now computed using exact arithmetic, as elements of a cyclotomic field. This is a change from Regina 4.96 and earlier, which computed floating-point approximations instead.
- Python
- This routine returns a Python dictionary. It also returns by value, not by reference (i.e., if more Turaev-Viro invariants are computed later on, the dictionary that was originally returned will not change as a result).
- Returns
- the cache of all Turaev-Viro invariants that have already been calculated.
- See also
- turaevViro
◆ anonID()
|
inherited |
A unique string ID that can be used in place of a packet ID.
This is an alternative to Packet::internalID(), and is designed for use when Held is not actually wrapped by a PacketOf<Held>. (An example of such a scenario is when a normal surface list needs to write its triangulation to file, but the triangulation is a standalone object that is not stored in a packet.)
The ID that is returned will:
- remain fixed throughout the lifetime of the program for a given object, even if the contents of the object are changed;
- not clash with the anonID() returned from any other object, or with the internalID() returned from any packet of any type;
These IDs are not preserved when copying or moving one object to another, and are not preserved when writing to a Regina data file and then reloading the file contents.
- Warning
- If this object is wrapped in a PacketOf<Held>, then anonID() and Packet::internalID() may return different values.
See Packet::internalID() for further details.
- Returns
- a unique ID that identifies this object.
◆ barycentricSubdivision()
|
inherited |
Does a barycentric subdivision of the triangulation.
This is done in-place, i.e., the triangulation will be modified directly.
Each top-dimensional simplex s is divided into (dim + 1) factorial sub-simplices by placing an extra vertex at the centroid of every face of every dimension. Each of these sub-simplices t is described by a permutation p of (0, ..., dim). The vertices of such a sub-simplex t are:
- vertex p[0] of s;
- the centre of edge (p[0], p[1]) of s;
- the centroid of triangle (p[0], p[1], p[2]) of s;
- ...
- the centroid of face (p[0], p[1], p[2], p[dim]) of s, which is the entire simplex s itself.
The sub-simplices have their vertices numbered in a way that mirrors the original simplex s:
- vertex p[0] of s will be labelled p[0] in t;
- the centre of edge (p[0], p[1]) of s will be labelled p[1] in t;
- the centroid of triangle (p[0], p[1], p[2]) of s will be labelled p[2] in t;
- ...
- the centroid of s itself will be labelled p[dim] in t.
In particular, if this triangulation is currently oriented, then this barycentric subdivision will preserve the orientation.
If simplex s has index i in the original triangulation, then its sub-simplex corresponding to permutation p will have index ((dim + 1)! * i + p.orderedSnIndex()) in the resulting triangulation. In other words: sub-simplices are ordered first according to the original simplex that contains them, and then according to the lexicographical ordering of the corresponding permutations p.
- Precondition
- dim is one of Regina's standard dimensions. This precondition is a safety net, since in higher dimensions the triangulation would explode too quickly in size (and for the highest dimensions, possibly beyond the limits of
size_t).
- Warning
- In dimensions 3 and 4, both the labelling and ordering of sub-simplices in the subdivided triangulation has changed as of Regina 5.1. (Earlier versions of Regina made no guarantee about the labelling and ordering; these guarantees are also new to Regina 5.1).
- Todo:
- Lock the topological properties of the underlying manifold, to avoid recomputing them after the subdivision. However, only do this for valid triangulations (since we can have scenarios where invalid triangulations becoming valid and ideal after subdivision, which may change properties such as Triangulation<4>::knownSimpleLinks).
◆ boundaryComponent()
|
inlineinherited |
Returns the requested boundary component of this triangulation.
Note that each time the triangulation changes, all boundary components will be deleted and replaced with new ones. Therefore this object should be considered temporary only.
- Parameters
-
index the index of the desired boundary component; this must be between 0 and countBoundaryComponents()-1 inclusive.
- Returns
- the requested boundary component.
◆ boundaryComponents()
|
inlineinherited |
Returns an object that allows iteration through and random access to all boundary components of this triangulation.
Note that, in Regina's standard dimensions, each ideal vertex forms its own boundary component, and some invalid vertices do also. See the BoundaryComponent class notes for full details on what constitutes a boundary component in standard and non-standard dimensions.
The object that is returned is lightweight, and can be happily copied by value. The C++ type of the object is subject to change, so C++ users should use auto (just like this declaration does).
The returned object is guaranteed to be an instance of ListView, which means it offers basic container-like functions and supports C++11 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:
The object that is returned will remain up-to-date and valid for as long as the triangulation exists. In contrast, however, remember that the individual boundary components within this list will be deleted and replaced each time the triangulation changes. Therefore it is best to treat this object as temporary only, and to call boundaryComponents() again each time you need it.
- Returns
- access to the list of all boundary components.
◆ boundaryMap() [1/2]
|
inherited |
Returns the boundary map from subdim-faces to (subdim-1)-faces of the triangulation.
This is the boundary map that you would use if you were building the homology groups manually from a chain complex.
Unlike homology(), this code does not use the dual skeleton: instead it uses the primal (i.e., ordinary) skeleton.
- The main advantage of this is that you can easily match rows and columns of the returned matrix to faces of this triangulation.
- The main disadvantage is that ideal vertices are not treated as though they were truncated; instead they are just treated as 0-faces that appear as part of the chain complex.
The matrix that is returned should be thought of as acting on column vectors. Specifically, the cth column of the matrix corresponds to the cth subdim-face of this triangulation, and the rth row corresponds to the rth (subdim-1)-face of this triangulation. All faces are oriented according to the permutations returned by Simplex::faceMapping(), or equivalently, by FaceEmbedding::vertices().
If you wish to convert these boundary maps to homology groups yourself, either the AbelianGroup class (if you do not need to track which face is which) or the MarkedAbelianGroup class (if you do need to track individual faces) can help you do this.
Note that, unlike many of the templated face-related routines, this routine explicitly supports the case subdim = dim.
- Precondition
- This triangulation is valid and non-empty.
- Python
- Not present, since Python does not support templates. Python users can instead use the variant
homologyMap(subdim).
- Template Parameters
-
subdim the face dimension; this must be between 1 and dim inclusive.
- Returns
- the boundary map from subdim-faces to (subdim-1)-faces.
◆ boundaryMap() [2/2]
|
inlineinherited |
Returns the boundary map from subdim-faces to (subdim-1)-faces of the triangulation, where the face dimension does not need to be known until runtime.
For C++ programmers who know subdim at compile time, you are better off using the template function boundaryMap<subdim>() instead, which is slightly faster.
See the templated boundaryMap<subdim>() for full details on what this function computes and how the matrix it returns should be interpreted.
- Precondition
- This triangulation is valid and non-empty.
- Exceptions
-
InvalidArgument the face dimension subdim is outside the supported range (i.e., less than 1 or greater than dim).
- Parameters
-
subdim the face dimension; this must be between 1 and dim inclusive.
- Returns
- the boundary map from subdim-faces to (subdim-1)-faces.
◆ calculatedSkeleton()
|
inlineprotectedinherited |
Determines whether the skeletal objects and properties of this triangulation have been calculated.
These are only calculated "on demand", when a skeletal property is first queried.
- Returns
trueif and only if the skeleton has been calculated.
◆ clearBaseProperties()
|
protectedinherited |
Clears all properties that are managed by this base class.
This includes deleting all skeletal objects and emptying the corresponding internal lists, as well as clearing other cached properties and deallocating the corresponding memory where required.
Note that TriangulationBase almost never calls this routine itself (the one exception is the copy assignment operator). Typically clearBaseProperties() is only ever called by Triangulation<dim>::clearAllProperties(), which in turn is called by "atomic" routines that change the triangluation (before firing packet change events), as well as the Triangulation<dim> destructor.
◆ closeBook()
| bool regina::Triangulation< 3 >::closeBook | ( | Edge< 3 > * | e, |
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a book closing move about the given boundary edge.
This involves taking a boundary edge of the triangulation and folding together the two boundary triangles on either side. This move is the inverse of the openBook() move, and is used to simplify the boundary of the triangulation. This move can be done if:
- the edge e is a boundary edge;
- the two vertices opposite e in the boundary triangles that contain e are valid and distinct;
- the boundary component containing e contains more than two triangles.
There are in fact several other distinctness conditions on the nearby edges and triangles, but they follow automatically from the conditions above.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this operation will (trivially) preserve the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument f) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given edge is an edge of this triangulation.
- Parameters
-
e the edge about which to perform the move. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ collapseEdge()
| bool regina::Triangulation< 3 >::collapseEdge | ( | Edge< 3 > * | e, |
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a collapse of an edge between two distinct vertices.
This operation (when it is allowed) does not change the topology of the manifold, decreases the number of vertices by one, and also decreases the number of tetrahedra.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If you are trying to reduce the number of vertices without changing the topology, and if e is an edge connecting an internal vertex with some different vertex, then either collapseEdge() or pinchEdge() may be more appropriate for your situation.
- The advantage of collapseEdge() is that it decreases the number of tetrahedra, whereas pinchEdge() increases this number (but only by two).
- The disadvantages of collapseEdge() are that it cannot always be performed, and its validity tests are expensive; pinchEdge() on the other hand can always be used for edges e of the type described above.
If this triangulation is currently oriented, then this operation will preserve the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.
The eligibility requirements for this move are somewhat involved, and are discussed in detail in the collapseEdge() source code for those who are interested.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given edge is an edge of this triangulation.
- Parameters
-
e the edge to collapse. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the given edge may be collapsed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ component()
|
inlineinherited |
Returns the requested connected component of this triangulation.
Note that each time the triangulation changes, all component objects will be deleted and replaced with new ones. Therefore this component object should be considered temporary only.
- Parameters
-
index the index of the desired component; this must be between 0 and countComponents()-1 inclusive.
- Returns
- the requested component.
◆ components()
|
inlineinherited |
Returns an object that allows iteration through and random access to all components of this triangulation.
The object that is returned is lightweight, and can be happily copied by value. The C++ type of the object is subject to change, so C++ users should use auto (just like this declaration does).
The returned object is guaranteed to be an instance of ListView, which means it offers basic container-like functions and supports C++11 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:
The object that is returned will remain up-to-date and valid for as long as the triangulation exists. In contrast, however, remember that the individual component objects within this list will be deleted and replaced each time the triangulation changes. Therefore it is best to treat this object as temporary only, and to call components() again each time you need it.
- Returns
- access to the list of all components.
◆ connectedSumWith()
| void regina::Triangulation< 3 >::connectedSumWith | ( | const Triangulation< 3 > & | other | ) |
Forms the connected sum of this triangulation with the given triangulation.
This triangulation will be altered directly.
If this and the given triangulation are both oriented, then the result will be oriented also, and the connected sum will respect these orientations.
If one or both triangulations contains multiple connected components, this routine will connect the components containing tetrahedron 0 of each triangulation, and will copy any additional components across with no modification.
If either triangulation is empty, then the result will simply be a clone of the other triangulation.
This and/or the given triangulation may be bounded or ideal, or even invalid; in all cases the connected sum will be formed correctly. Note, however, that the result might possibly contain internal vertices (even if the original triangulations do not).
It is allowed to pass this triangulation as other.
- Parameters
-
other the triangulation to sum with this.
◆ countBoundaryComponents()
|
inlineinherited |
Returns the number of boundary components in this triangulation.
Note that, in Regina's standard dimensions, each ideal vertex forms its own boundary component, and some invalid vertices do also. See the BoundaryComponent class notes for full details on what constitutes a boundary component in standard and non-standard dimensions.
- Returns
- the number of boundary components.
◆ countBoundaryFacets()
|
inlineinherited |
Returns the total number of boundary facets in this triangulation.
This routine counts facets of top-dimensional simplices that are not glued to some adjacent top-dimensional simplex.
- Returns
- the total number of boundary facets.
◆ countBoundaryTriangles()
|
inline |
A dimension-specific alias for countBoundaryFacets().
See countBoundaryFacets() for further information.
◆ countComponents()
|
inlineinherited |
Returns the number of connected components in this triangulation.
- Returns
- the number of connected components.
◆ countEdges()
|
inlineinherited |
A dimension-specific alias for countFaces<1>().
This alias is available for all dimensions dim.
See countFaces() for further information.
◆ countFaces() [1/2]
|
inlineinherited |
Returns the number of subdim-faces in this triangulation.
This is the fastest way to count faces if you know subdim at compile time.
For convenience, this routine explicitly supports the case subdim = dim. This is not the case for the routines face() and faces(), which give access to individual faces (the reason relates to the fact that top-dimensional simplices are built manually, whereas lower-dimensional faces are deduced properties).
- Python
- Not present, since Python does not support templates. Python users can instead use the variant
countFaces(subdim).
- Template Parameters
-
subdim the face dimension; this must be between 0 and dim inclusive.
- Returns
- the number of subdim-faces.
◆ countFaces() [2/2]
|
inlineinherited |
Returns the number of subdim-faces in this triangulation, where the face dimension does not need to be known until runtime.
This routine takes linear time in the dimension dim. For C++ programmers who know subdim at compile time, you are better off using the template function countFaces<subdim>() instead, which is fast constant time.
For convenience, this routine explicitly supports the case subdim = dim. This is not the case for the routines face() and faces(), which give access to individual faces (the reason relates to the fact that top-dimensional simplices are built manually, whereas lower-dimensional faces are deduced properties).
- Exceptions
-
InvalidArgument the face dimension subdim is outside the supported range (i.e., negative or greater than dim).
- Parameters
-
subdim the face dimension; this must be between 0 and dim inclusive.
- Returns
- the number of subdim-faces.
◆ countPentachora()
|
inlineinherited |
A dimension-specific alias for countFaces<4>().
This alias is available for dimensions dim ≥ 4.
See countFaces() for further information.
◆ countTetrahedra()
|
inlineinherited |
A dimension-specific alias for countFaces<3>().
This alias is available for dimensions dim ≥ 3.
See countFaces() for further information.
◆ countTriangles()
|
inlineinherited |
A dimension-specific alias for countFaces<2>().
This alias is available for all dimensions dim.
See countFaces() for further information.
◆ countVertices()
|
inlineinherited |
A dimension-specific alias for countFaces<0>().
This alias is available for all dimensions dim.
See countFaces() for further information.
◆ dehydrate()
| std::string regina::Triangulation< 3 >::dehydrate | ( | ) | const |
Dehydrates this triangulation into an alphabetical string.
A dehydration string is a compact text representation of a triangulation, introduced by Callahan, Hildebrand and Weeks for their cusped hyperbolic census (see below). The dehydration string of an n-tetrahedron triangulation consists of approximately (but not precisely) 5n/2 lower-case letters.
Dehydration strings come with some restrictions:
- They rely on the triangulation being "canonical" in some combinatorial sense. This is not enforced here; instead a combinatorial isomorphism is applied to make the triangulation canonical, and this isomorphic triangulation is dehydrated instead. Note that the original triangulation is not changed.
- They require the triangulation to be connected.
- They require the triangulation to have no boundary triangles (though ideal triangulations are fine).
- They can only support triangulations with at most 25 tetrahedra.
The routine rehydrate() can be used to recover a triangulation from a dehydration string. Note that the triangulation recovered might not be identical to the original, but it is guaranteed to be an isomorphic copy.
For a full description of the dehydrated triangulation format, see A Census of Cusped Hyperbolic 3-Manifolds, Callahan, Hildebrand and Weeks, Mathematics of Computation 68/225, 1999.
- Exceptions
-
NotImplemented Either this triangulation is disconnected, it has boundary triangles, or it contains more than 25 tetrahedra.
- Returns
- a dehydrated representation of this triangulation (or an isomorphic variant of this triangulation).
◆ detail()
|
inherited |
Returns a detailed text representation of this object.
This text may span many lines, and should provide the user with all the information they could want. It should be human-readable, should not contain extremely long lines (which cause problems for users reading the output in a terminal), and should end with a final newline. There are no restrictions on the underlying character set.
- Returns
- a detailed text representation of this object.
◆ dumpConstruction()
|
inherited |
Returns C++ code that can be used to reconstruct this triangulation.
This code will call Triangulation<dim>::fromGluings(), passing a hard-coded C++11 initialiser list.
The main purpose of this routine is to generate this hard-coded initialiser list, which can be tedious and error-prone to write by hand.
Note that the number of lines of code produced grows linearly with the number of simplices. If this triangulation is very large, the returned string will be very large as well.
- Returns
- the C++ code that was generated.
◆ edge()
|
inlineinherited |
◆ edges()
|
inlineinherited |
A dimension-specific alias for faces<1>().
This alias is available for all dimensions dim.
See faces() for further information.
◆ ensureSkeleton()
|
inlineprotectedinherited |
Ensures that all "on demand" skeletal objects have been calculated.
◆ enterTextTriangulation()
|
static |
Deprecated function that allows the user to interactively enter a triangulation in plain text.
Prompts will be sent to the given output stream and information will be read from the given input stream.
- Deprecated:
- This is essentially an ancient piece of an interactive user interface, buried inside what is otherwise a mathematical library, and dating back to before Regina offered Python bindings. Nowadays there are better ways of interacting with Regina at a low level (e.g., Python), and so this routine will soon be removed completely.
- Python
- This routine takes no parameters; in and out are always assumed to be standard input and standard output respectively.
- Parameters
-
in the input stream from which text will be read. out the output stream to which prompts will be written.
- Returns
- the triangulation entered in by the user.
◆ eulerCharManifold()
| long regina::Triangulation< 3 >::eulerCharManifold | ( | ) | const |
Returns the Euler characteristic of the corresponding compact 3-manifold.
Instead of simply calculating V-E+F-T, this routine also:
- treats ideal vertices as surface boundary components (i.e., effectively truncates them);
- truncates invalid boundary vertices (i.e., boundary vertices whose links are not discs);
- truncates the projective plane cusps at the midpoints of invalid edges (edges identified with themselves in reverse).
For ideal triangulations, this routine therefore computes the proper Euler characteristic of the manifold (unlike eulerCharTri(), which does not).
For triangulations whose vertex links are all spheres or discs, this routine and eulerCharTri() give identical results.
- Returns
- the Euler characteristic of the corresponding compact manifold.
◆ eulerCharTri()
|
inlineinherited |
Returns the Euler characteristic of this triangulation.
This will be evaluated strictly as the alternating sum of the number of i-faces (that is, countVertices() - countEdges() + countTriangles() - ...).
Note that this routine handles ideal triangulations in a non-standard way. Since it computes the Euler characteristic of the triangulation (and not the underlying manifold), this routine will treat each ideal boundary component as a single vertex, and not as an entire (dim-1)-dimensional boundary component.
In Regina's standard dimensions, for a routine that handles ideal boundary components properly (by treating them as (dim-1)-dimensional boundary components when computing Euler characteristic), you can use the routine eulerCharManifold() instead.
- Returns
- the Euler characteristic of this triangulation.
◆ face() [1/2]
|
inlineinherited |
Returns the requested subdim-face of this triangulation, in a way that is optimised for Python programmers.
For C++ users, this routine is not very useful: since precise types must be know at compile time, this routine returns a std::variant v that could store a pointer to any class Face<dim, ...>. This means you cannot access the face directly: you will still need some kind of compile-time knowledge of subdim before you can extract and use an appropriate Face<dim, subdim> object from v. However, once you know subdim at compile time, you are better off using the (simpler and faster) routine face<subdim>() instead.
For Python users, this routine is much more useful: the return type can be chosen at runtime, and so this routine simply returns a Face<dim, subdim> object of the appropriate face dimension that you can use immediately.
The specific return type for C++ programmers will be std::variant<Face<dim, 0>*, ..., Face<dim, dim-1>*>.
- Exceptions
-
InvalidArgument the face dimension subdim is outside the supported range (i.e., negative, or greater than or equal to dim).
- Parameters
-
subdim the face dimension; this must be between 0 and dim-1 inclusive. index the index of the desired face, ranging from 0 to countFaces<subdim>()-1 inclusive.
- Returns
- the requested face.
◆ face() [2/2]
|
inlineinherited |
Returns the requested subdim-face of this triangulation, in a way that is optimised for C++ programmers.
- Python
- Not present, since Python does not support templates. Python users can instead use the variant
face(subdim, index).
- Template Parameters
-
subdim the face dimension; this must be between 0 and dim-1 inclusive.
- Parameters
-
index the index of the desired face, ranging from 0 to countFaces<subdim>()-1 inclusive.
- Returns
- the requested face.
◆ faces() [1/2]
|
inlineinherited |
Returns an object that allows iteration through and random access to all subdim-faces of this triangulation, in a way that is optimised for C++ programmers.
The object that is returned is lightweight, and can be happily copied by value. The C++ type of the object is subject to change, so C++ users should use auto (just like this declaration does).
The returned object is guaranteed to be an instance of ListView, which means it offers basic container-like functions and supports C++11 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:
The object that is returned will remain up-to-date and valid for as long as the triangulation exists. In contrast, however, remember that the individual faces within this list will be deleted and replaced each time the triangulation changes. Therefore it is best to treat this object as temporary only, and to call faces() again each time you need it.
- Python
- Not present, since Python does not support templates. Python users can instead use the variant
faces(subdim).
- Template Parameters
-
subdim the face dimension; this must be between 0 and dim-1 inclusive.
- Returns
- access to the list of all subdim-faces.
◆ faces() [2/2]
|
inlineinherited |
Returns an object that allows iteration through and random access to all subdim-faces of this triangulation, in a way that is optimised for Python programmers.
C++ users should not use this routine. The return type must be fixed at compile time, and so it is a std::variant that can hold any of the lightweight return types from the templated faces<subdim>() function. This means that the return value will still need compile-time knowledge of subdim to extract and use the appropriate face objects. However, once you know subdim at compile time, you are much better off using the (simpler and faster) routine faces<subdim>() instead.
For Python users, this routine is much more useful: the return type can be chosen at runtime, and so this routine returns a Python list of Face<dim, subdim> objects (holding all the subdim-faces of the triangulation), which you can use immediately.
- Exceptions
-
InvalidArgument the face dimension subdim is outside the supported range (i.e., negative, or greater than or equal to dim).
- Parameters
-
subdim the face dimension; this must be between 0 and dim-1 inclusive.
- Returns
- access to the list of all subdim-faces.
◆ fillTorus() [1/2]
| bool regina::Triangulation< 3 >::fillTorus | ( | Edge< 3 > * | e0, |
| Edge< 3 > * | e1, | ||
| Edge< 3 > * | e2, | ||
| unsigned long | cuts0, | ||
| unsigned long | cuts1, | ||
| unsigned long | cuts2 | ||
| ) |
Fills a two-triangle torus boundary component by attaching a solid torus along a given curve.
The three edges of the boundary component should be passed as the arguments e0, e1 and e2. The boundary component will then be filled with a solid torus whose meridional curve cuts these three edges cuts0, cuts1 and cuts2 times respectively.
For the filling to be performed successfully, the three given edges must belong to the same boundary component, and this boundary component must be a two-triangle torus. Moreover, the integers cuts0, cuts1 and cuts2 must be coprime, and two of them must add to give the third. If any of these conditions are not met, then this routine will do nothing and return false.
The triangulation will be simplified before returning.
There are two versions of fillTorus(); the other takes a boundary component, and sets e0, e1 and e2 to its three edges according to Regina's own edge numbering. This version of fillTorus() should be used when you know how the filling curve cuts each boundary edge but you do not know how these edges are indexed in the corresponding boundary component.
- Parameters
-
e0 one of the three edges of the boundary component to fill. e1 the second of the three edges of the boundary component to fill. e2 the second of the three edges of the boundary component to fill. cuts0 the number of times that the meridional curve of the new solid torus should cut the edge e0. cuts1 the number of times that the meridional curve of the new solid torus should cut the edge e1. cuts2 the number of times that the meridional curve of the new solid torus should cut the edge e2.
- Returns
trueif the boundary component was filled successfully, orfalseif one of the required conditions as described above is not satisfied.
◆ fillTorus() [2/2]
| bool regina::Triangulation< 3 >::fillTorus | ( | unsigned long | cuts0, |
| unsigned long | cuts1, | ||
| unsigned long | cuts2, | ||
| BoundaryComponent< 3 > * | bc = nullptr |
||
| ) |
Fills a two-triangle torus boundary component by attaching a solid torus along a given curve.
The boundary component to be filled should be passed as the argument bc; if the triangulation has exactly one boundary component then you may omit bc (i.e., pass null), and the (unique) boundary component will be inferred.
If the boundary component cannot be inferred, and/or if the selected boundary component is not a two-triangle torus, then this routine will do nothing and return false.
Otherwise the given boundary component will be filled with a solid torus whose meridional curve cuts the edges bc->edge(0), bc->edge(1) and bc->edge(2) a total of cuts0, cuts1 and cuts2 times respectively.
For the filling to be performed successfully, the integers cuts0, cuts1 and cuts2 must be coprime, and two of them must add to give the third. Otherwise, as above, this routine will do nothing and return false.
The triangulation will be simplified before returning.
There are two versions of fillTorus(); the other takes three explicit edges instead of a boundary component. You should use the other version if you know how the filling curve cuts each boundary edge but you do not know how these edges are indexed in the boundary component.
- Parameters
-
cuts0 the number of times that the meridional curve of the new solid torus should cut the edge bc->edge(0).cuts1 the number of times that the meridional curve of the new solid torus should cut the edge bc->edge(1).cuts2 the number of times that the meridional curve of the new solid torus should cut the edge bc->edge(2).bc the boundary component to fill. If the triangulation has precisely one boundary component then this may be null.
- Returns
trueif the boundary component was filled successfully, orfalseif one of the required conditions as described above is not satisfied.
◆ findAllIsomorphisms()
|
inlineinherited |
Finds all ways in which this triangulation is combinatorially isomorphic to the given triangulation.
This routine behaves identically to isIsomorphicTo(), except that instead of returning just one isomorphism, all such isomorphisms will be found and processed. See the isIsomorphicTo() notes for details on this.
For each isomorphism that is found, this routine will call action (which must be a function or some other callable object).
- The first argument to action must be of type
(const Isomorphism<dim>&); this will be a reference to the isomorphism that was found. If action wishes to keep the isomorphism, it should take a deep copy (not a reference), since the isomorphism may be changed and reused after action returns. - If there are any additional arguments supplied in the list args, then these will be passed as subsequent arguments to action.
- action must return a
bool. A return value offalseindicates that the search for isomorphisms should continue, and a return value oftrueindicates that the search should terminate immediately. - This triangulation must remain constant while the search runs (i.e., action must not modify the triangulation).
- Warning
- For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
- Python
- There are two versions of this function available in Python. The first form is
findAllIsomorphisms(other, action), which mirrors the C++ function: it takes action which may be a pure Python function, the return value indicates whether action ever terminated the search, but it does not take an additonal argument list (args). The second form isfindAllIsomorphisms(other), which returns a Python list containing all of the isomorphisms that were found.
- Parameters
-
other the triangulation to compare with this one. action a function (or other callable object) to call for each isomorphism that is found. args any additional arguments that should be passed to action, following the initial isomorphism argument.
- Returns
trueif action ever terminated the search by returningtrue, orfalseif the search was allowed to run to completion.
◆ findAllSubcomplexesIn()
|
inlineinherited |
Finds all ways in which an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components).
This routine behaves identically to isContainedIn(), except that instead of returning just one isomorphism (which may be boundary incomplete and need not be onto), all such isomorphisms will be found and processed. See the isContainedIn() notes for details on this.
For each isomorphism that is found, this routine will call action (which must be a function or some other callable object).
- The first argument to action must be of type
(const Isomorphism<dim>&); this will be a reference to the isomorphism that was found. If action wishes to keep the isomorphism, it should take a deep copy (not a reference), since the isomorphism may be changed and reused after action returns. - If there are any additional arguments supplied in the list args, then these will be passed as subsequent arguments to action.
- action must return a
bool. A return value offalseindicates that the search for isomorphisms should continue, and a return value oftrueindicates that the search should terminate immediately. - This triangulation must remain constant while the search runs (i.e., action must not modify the triangulation).
- Warning
- For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
- Python
- There are two versions of this function available in Python. The first form is
findAllSubcomplexesIn(other, action), which mirrors the C++ function: it takes action which may be a pure Python function, the return value indicates whether action ever terminated the search, but it does not take an additonal argument list (args). The second form isfindAllSubcomplexesIn(other), which returns a Python list containing all of the isomorphisms that were found.
- Parameters
-
other the triangulation in which to search for isomorphic copies of this triangulation. action a function (or other callable object) to call for each isomorphism that is found. args any additional arguments that should be passed to action, following the initial isomorphism argument.
- Returns
trueif action ever terminated the search by returningtrue, orfalseif the search was allowed to run to completion.
◆ findStrictAngleStructure()
|
inline |
A deprecated alias for strictAngleStructure(), which returns a strict angle structure on this triangulation if one exists.
- Deprecated:
- This routine has been renamed to strictAngleStructure(). See that routine for further details.
- Exceptions
-
NoSolution no strict angle structure exists on this triangulation.
- Returns
- a strict angle structure on this triangulation, if one exists.
◆ finiteToIdeal()
|
inherited |
Converts each real boundary component into a cusp (i.e., an ideal vertex).
Only boundary components formed from real (dim-1)-faces will be affected; ideal boundary components are already cusps and so will not be changed.
One side-effect of this operation is that all spherical boundary components will be filled in with balls.
This operation is performed by attaching a new dim-simplex to each boundary (dim-1)-face, and then gluing these new simplices together in a way that mirrors the adjacencies of the underlying boundary facets. Each boundary component will thereby be pushed up through the new simplices and converted into a cusp formed using vertices of these new simplices.
In Regina's standard dimensions, where triangulations also support an idealToFinite() operation, this routine is a loose converse of that operation.
In dimension 2, every boundary component is spherical and so this routine simply fills all the punctures in the underlying surface. (In dimension 2, triangulations cannot have cusps).
- Warning
- If a real boundary component contains vertices whose links are not discs, this operation may have unexpected results.
- Returns
trueif changes were made, orfalseif the original triangulation contained no real boundary components.
◆ fourFourMove()
| bool regina::Triangulation< 3 >::fourFourMove | ( | Edge< 3 > * | e, |
| int | newAxis, | ||
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a 4-4 move about the given edge.
This involves replacing the four tetrahedra joined at that edge with four tetrahedra joined along a different edge. Consider the octahedron made up of the four original tetrahedra; this has three internal axes. The initial four tetrahedra meet along the given edge which forms one of these axes; the new tetrahedra will meet along a different axis. This move can be done iff (i) the edge is valid and non-boundary, and (ii) the four tetrahedra are distinct.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this 4-4 move will label the new tetrahedra in a way that preserves the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given edge is an edge of this triangulation.
- Parameters
-
e the edge about which to perform the move. newAxis Specifies which axis of the octahedron the new tetrahedra should meet along; this should be 0 or 1. Consider the four original tetrahedra in the order described by Edge<3>::embedding(0,...,3); call these tetrahedra 0, 1, 2 and - If newAxis is 0, the new axis will separate tetrahedra 0 and 1 from 2 and 3. If newAxis is 1, the new axis will separate tetrahedra 1 and 2 from 3 and 0.
check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ fromGluings() [1/2]
|
staticinherited |
Creates a triangulation from a list of gluings.
This routine is an analogue to the variant of fromGluings() that takes a C++11 initialiser list; however, here the input data may be constructed at runtime (which makes it accessible to Python, amongst other things).
The iterator range (beginGluings, endGluings) should encode the list of gluings for the triangulation. Each iterator in this range must dereference to a tuple of the form (simp, facet, adj, gluing); here simp, facet and adj are all integers, and gluing is of type Perm<dim+1>. Each such tuple indicates that facet facet of top-dimensional simplex number simp should be glued to top-dimensional simplex number adj using the permutation gluing. In other words, such a tuple encodes the same information as calling simplex(simp).join(facet, simplex(adj), gluing) upon the triangulation being constructed.
Every gluing should be encoded from one direction only. This means, for example, that to build a closed 3-manifold triangulation with n tetrahedra, you would pass a list of 2n such tuples. If you attempt to make the same gluing twice (e.g., once from each direction), then this routine will throw an exception.
Any facet of a simplex that does not feature in the given list of gluings (either as a source or a destination) will be left as a boundary facet.
Note that, as usual, the top-dimensional simplices are numbered 0,...,(size-1), and the facets of each simplex are numbered 0,...,dim.
As an example, Python users can construct the figure eight knot complement as follows:
- Note
- The assumption is that the iterators dereference to a std::tuple<size_t, int, size_t, Perm<dim+1>>. However, this is not strictly necessary - the dereferenced type may be any type that supports std::get (and for which std::get<0..3>() yields suitable integer/permutation types).
- Exceptions
-
InvalidArgument the given list of gluings does not correctly describe a triangulation with size top-dimensional simplices.
- Python
- The gluings should be passed as a single Python list of tuples (not an iterator pair).
- Parameters
-
size the number of top-dimensional simplices in the triangulation to construct. beginGluings the beginning of the list of gluings, as described above. endGluings a past-the-end iterator indicating the end of the list of gluings.
- Returns
- the reconstructed triangulation.
◆ fromGluings() [2/2]
|
inlinestaticinherited |
Creates a triangulation from a hard-coded list of gluings.
This routine takes a C++11 initialiser list, which makes it useful for creating hard-coded examples directly in C++ code without needing to write a tedious sequence of calls to Simplex<dim>::join().
Each element of the initialiser list should be a tuple of the form (simp, facet, adj, gluing), which indicates that facet facet of top-dimensional simplex number simp should be glued to top-dimensional simplex number adj using the permutation gluing. In other words, such a tuple encodes the same information as calling simplex(simp).join(facet, simplex(adj), gluing) upon the triangulation being constructed.
Every gluing should be encoded from one direction only. This means, for example, that to build a closed 3-manifold triangulation with n tetrahedra, you would pass a list of 2n such tuples. If you attempt to make the same gluing twice (e.g., once from each direction), then this routine will throw an exception.
Any facet of a simplex that does not feature in the given list of gluings (either as a source or a destination) will be left as a boundary facet.
Note that, as usual, the top-dimensional simplices are numbered 0,...,(size-1), and the facets of each simplex are numbered 0,...,dim.
As an example, you can construct the figure eight knot complement using the following code:
- Note
- If you have an existing triangulation that you would like to hard-code in this way, you can call dumpConstruction() to generate the corresponding C++ source code.
- Exceptions
-
InvalidArgument the given list of gluings does not correctly describe a triangulation with size top-dimensional simplices.
- Python
- Not available, but there is a variant of fromGluings() that takes the same data using a Python list (which need not be constant).
- Parameters
-
size the number of top-dimensional simplices in the triangulation to construct. gluings describes the gluings between these top-dimensional simplices, as described above.
- Returns
- the reconstructed triangulation.
◆ fromIsoSig()
|
staticinherited |
Recovers a full triangulation from an isomorphism signature.
See isoSig() for more information on isomorphism signatures. It will be assumed that the signature describes a triangulation of dimension dim.
Currently this routine only supports isomorphism signatures that were created with the default encoding (i.e., there was no Encoding template parameter passed to isoSig()).
Calling isoSig() followed by fromIsoSig() is not guaranteed to produce an identical triangulation to the original, but it is guaranteed to produce a combinatorially isomorphic triangulation. In other words, fromIsoSig() may reconstruct the triangulation with its simplices and/or vertices relabelled. The optional argument to isoSig() allows you to determine the precise relabelling that will be used, if you need to know it.
For a full and precise description of the isomorphism signature format for 3-manifold triangulations, see Simplification paths in the Pachner graphs of closed orientable 3-manifold triangulations, Burton, 2011, arXiv:1110.6080. The format for other dimensions is essentially the same, but with minor dimension-specific adjustments.
- Warning
- Do not mix isomorphism signatures between dimensions! It is possible that the same string could corresponding to both a p-dimensional triangulation and a q-dimensional triangulation for different dimensions p and q.
- Exceptions
-
InvalidArgument the given string was not a valid dim-dimensional isomorphism signature created using the default encoding.
- Parameters
-
sig the isomorphism signature of the triangulation to construct. Note that isomorphism signatures are case-sensitive (unlike, for example, dehydration strings for 3-manifolds).
- Returns
- the reconstructed triangulation.
◆ fromSig()
|
inlinestaticinherited |
Alias for fromIsoSig(), to recover a full triangulation from an isomorphism signature.
This alias fromSig() is provided to assist with generic code that can work with both triangulations and links.
See fromIsoSig() for further details.
- Exceptions
-
InvalidArgument the given string was not a valid dim-dimensional isomorphism signature created using the default encoding.
- Parameters
-
sig the isomorphism signature of the triangulation to construct. Note that isomorphism signatures are case-sensitive (unlike, for example, dehydration strings for 3-manifolds).
- Returns
- the reconstructed triangulation.
◆ fromSnapPea()
|
static |
Extracts the tetrahedron gluings from a string that contains the full contents of a SnapPea data file.
All other SnapPea-specific information (such as peripheral curves, and the manifold name) will be ignored, since Regina's Triangulation<3> class does not track such information itself.
If you wish to preserve all SnapPea-specific information from the data file, you should work with the SnapPeaTriangulation class instead (which uses the SnapPea kernel directly, and can therefore store anything that SnapPea can).
If you wish to read a triangulation from a SnapPea file, you should likewise call the SnapPeaTriangulation constructor, giving the filename as argument. This will read all SnapPea-specific information (as described above), and also avoids constructing an enormous intermediate string.
- Warning
- This routine is "lossy", in that drops SnapPea-specific information (as described above). Unless you specifically need an Triangulation<3> (not an SnapPeaTriangulation) or you need to avoid calling routines from the SnapPea kernel, it is highly recommended that you create a SnapPeaTriangulation from the given file contents instead. See the string-based SnapPeaTriangulation constructor for how to do this.
- Exceptions
-
InvalidArgument the given SnapPea data was not in the correct format.
- Parameters
-
snapPeaData a string containing the full contents of a SnapPea data file.
- Returns
- a native Regina triangulation extracted from the given SnapPea data.
◆ fundamentalGroup()
|
inherited |
Returns the fundamental group of this triangulation.
The fundamental group is computed in the dual 2-skeleton. This means:
- If the triangulation contains any ideal vertices, the fundamental group will be calculated as if each such vertex had been truncated.
- Likewise, if the triangulation contains any invalid faces of dimension 0,1,...,(dim-3), these will effectively be truncated also.
- In contrast, if the triangulation contains any invalid (dim-2)-faces (i.e., codimension-2-faces that are identified with themselves under a non-trivial map), the fundamental group will be computed without truncating the centroid of the face. For instance, if a 3-manifold triangulation has an edge identified with itself in reverse, then the fundamental group will be computed without truncating the resulting projective plane cusp. This means that, if a barycentric subdivision is performed on a such a triangulation, the result of fundamentalGroup() might change.
Bear in mind that each time the triangulation changes, the fundamental group will be deleted. Thus the reference that is returned from this routine should not be kept for later use. Instead, fundamentalGroup() should be called again; this will be instantaneous if the group has already been calculated.
- Precondition
- This triangulation has at most one component.
- Warning
- In dimension 3, if you are calling this from the subclass SnapPeaTriangulation then any fillings on the cusps will be ignored. (This is the same as for every routine implemented by Regina's Triangulation<3> class.) If you wish to compute the fundamental group with fillings, call SnapPeaTriangulation::fundamentalGroupFilled() instead.
- Returns
- the fundamental group.
◆ fVector()
|
inlineinherited |
Returns the f-vector of this triangulation, which counts the number of faces of all dimensions.
The vector that is returned will have length dim+1. If this vector is f, then f[k] will be the number of k-faces for each 0 ≤ k ≤ dim.
This routine is significantly more heavyweight than countFaces(). Its advantage is that, unlike the templatised countFaces(), it allows you to count faces whose dimensions are not known until runtime.
- Returns
- the f-vector of this triangulation.
◆ generalAngleStructure()
|
inline |
Returns a generalised angle structure on this triangulation, if one exists.
A generalised angle structure must satisfy the same matching equations as all angle structures do, but there is no constraint on the signs of the angles; in particular, negative angles are allowed. If a generalised angle structure does exist, then this routine is guaranteed to return one.
This routine is designed for scenarios where you already know that a generalised angle structure exists. This means:
- If no generalised angle structure exists, this routine will throw an exception, which will incur a significant overhead.
- It should be rare that you do not know in advance whether a generalised angle structure exists (see the simple conditions in the note below). However, if you don't yet know, you should call hasGeneralAngleStructure() first. If the answer is no, this will avoid the overhead of throwing and catching exceptions. If the answer is yes, this will have the side-effect of caching the angle structure, which means your subsequent call to generalAngleStructure() will be essentially instantaneous.
The underlying algorithm simply solves a system of linear equations, and so should be fast even for large triangulations.
The result of this routine is cached internally: as long as the triangulation does not change, multiple calls to generalAngleStructure() will return identical angle structures, and every call after the first be essentially instantaneous.
If the triangulation does change, however, then the cached angle structure will be deleted, and any reference that was returned before will become invalid.
- Note
- For a valid triangulation with no boundary faces, a generalised angle structure exists if and only if every vertex link is a torus or Klein bottle. The "only if" direction is a simple Euler characteristic calculation; for the "if" direction see "Angle structures and normal surfaces", Feng Luo and Stephan Tillmann, Trans. Amer. Math. Soc. 360:6 (2008), pp. 2849-2866).
- Exceptions
-
NoSolution no generalised angle structure exists on this triangulation.
- Returns
- a generalised angle structure on this triangulation, if one exists.
◆ hasBoundaryFacets()
|
inlineinherited |
Determines if this triangulation has any boundary facets.
This routine returns true if and only if the triangulation contains some top-dimension simplex with at least one facet that is not glued to an adjacent simplex.
- Returns
trueif and only if there are boundary facets.
◆ hasBoundaryTriangles()
|
inline |
A dimension-specific alias for hasBoundaryFacets().
See hasBoundaryFacets() for further information.
◆ hasCompressingDisc()
| bool regina::Triangulation< 3 >::hasCompressingDisc | ( | ) | const |
Searches for a compressing disc within the underlying 3-manifold.
Let M be the underlying 3-manifold and let B be its boundary. By a compressing disc, we mean a disc D properly embedded in M, where the boundary of D lies in B but does not bound a disc in B.
This routine will first call the heuristic routine hasSimpleCompressingDisc() in the hope of obtaining a fast answer. If this fails, it will do one of two things:
- If the triangulation is orientable and 1-vertex, it will use the linear programming and crushing machinery outlined in "Computing closed essential surfaces in knot complements", Burton, Coward and Tillmann, SCG '13, p405-414, 2013. This is often extremely fast, even for triangulations with many tetrahedra.
- If the triangulation is non-orientable or has multiple vertices then it will run a full enumeration of vertex normal surfaces, as described in "Algorithms for the complete decomposition of a closed 3-manifold", Jaco and Tollefson, Illinois J. Math. 39 (1995), 358-406. As the number of tetrahedra grows, this can become extremely slow.
This routine will work on a copy of this triangulation, not the original. In particular, the copy will be simplified, which means that there is no harm in calling this routine on an unsimplified triangulation.
If this triangulation has no boundary components, this routine will simply return false.
- Precondition
- This triangulation is valid and is not ideal.
- The underlying 3-manifold is irreducible.
- Warning
- This routine can be infeasibly slow for large triangulations (particularly those that are non-orientable or have multiple vertices), since it may need to perform a full enumeration of vertex normal surfaces, and since it might perform "large" operations on these surfaces such as cutting along them. See hasSimpleCompressingDisc() for a "heuristic shortcut" that is faster but might not give a definitive answer.
- Returns
trueif the underlying 3-manifold contains a compressing disc, orfalseif it does not.
◆ hasGeneralAngleStructure()
| bool regina::Triangulation< 3 >::hasGeneralAngleStructure | ( | ) | const |
Determines whether this triangulation supports a generalised angle structure.
A generalised angle structure must satisfy the same matching equations as all angle structures do, but there is no constraint on the signs of the angles; in particular, negative angles are allowed.
This routine returns false if and only if generalAngleStructure() throws an exception. However, if you do not know whether a generalised angle structure exists, then this routine is faster:
- If there is no generalised angle structure, this routine will avoid the overhead of throwing and catching exceptions.
- If there is a generalised angle structure, this routine will find and cache this angle structure, which means that any subsequent call to generalAngleStructure() to retrieve its details will be essentially instantaneous.
The underlying algorithm simply solves a system of linear equations, and so should be fast even for large triangulations.
- Note
- For a valid triangulation with no boundary faces, a generalised angle structure exists if and only if every vertex link is a torus or Klein bottle. The "only if" direction is a simple Euler characteristic calculation; for the "if" direction see "Angle structures and normal surfaces", Feng Luo and Stephan Tillmann, Trans. Amer. Math. Soc. 360:6 (2008), pp. 2849-2866).
- Even if the condition above is true and it is clear that a generalised angle structure should exist, this routine will still do the extra work to compute an explicit solution (in order to fulfil the promise made in the generalAngleStructure() documentation).
- Returns
trueif and only if a generalised angle structure exists on this triangulation.
◆ hasNegativeIdealBoundaryComponents()
|
inline |
Determines if this triangulation contains any ideal boundary components with negative Euler characteristic.
- Returns
trueif and only if there is at least one such boundary component.
◆ hasNonTrivialSphereOrDisc()
|
inline |
A deprecated alias for nonTrivialSphereOrDisc(), which searches for a non-vertex-linking normal sphere or disc within this triangulation.
- Deprecated:
- This routine has been renamed to nonTrivialSphereOrDisc(). See that routine for further details.
- Returns
- a non-vertex-linking normal sphere or disc, or no value if none exists.
◆ hasOctagonalAlmostNormalSphere()
|
inline |
A deprecated alias for octagonalAlmostNormalSphere(), which searches for an octagonal almost normal 2-sphere within this triangulation.
- Deprecated:
- This routine has been renamed to octagonalAlmostNormalSphere(). See that routine for further details.
- Precondition
- This triangulation is valid, closed, orientable, connected, and 0-efficient.
- Returns
- an octagonal almost normal 2-sphere, or no value if none exists.
◆ hasSimpleCompressingDisc()
| bool regina::Triangulation< 3 >::hasSimpleCompressingDisc | ( | ) | const |
Searches for a "simple" compressing disc inside this triangulation.
Let M be the underlying 3-manifold and let B be its boundary. By a compressing disc, we mean a disc D properly embedded in M, where the boundary of D lies in B but does not bound a disc in B.
By a simple compressing disc, we mean a compressing disc that has a very simple combinatorial structure (here "simple" is subject to change; see the warning below). Examples include the compressing disc inside a 1-tetrahedron solid torus, or a compressing disc formed from a single internal triangle surrounded by three boundary edges.
The purpose of this routine is to avoid working with normal surfaces within a large triangulation where possible. This routine is relatively fast, and if it returns true then this 3-manifold definitely contains a compressing disc. If this routine returns false then there might or might not be a compressing disc; the user will need to perform a full normal surface enumeration using hasCompressingDisc() to be sure.
This routine will work on a copy of this triangulation, not the original. In particular, the copy will be simplified, which means that there is no harm in calling this routine on an unsimplified triangulation.
If this triangulation has no boundary components, this routine will simply return false.
For further information on this test, see "The Weber-Seifert dodecahedral space is non-Haken", Benjamin A. Burton, J. Hyam Rubinstein and Stephan Tillmann, Trans. Amer. Math. Soc. 364:2 (2012), pp. 911-932.
- Warning
- The definition of "simple" is subject to change in future releases of Regina. That is, this routine may be expanded over time to identify more types of compressing discs (thus making it more useful as a "heuristic shortcut").
- Precondition
- This triangulation is valid and is not ideal.
- Returns
trueif a simple compressing disc was found, orfalseif not. Note that even with a return value offalse, there might still be a compressing disc (just not one with a simple combinatorial structure).
◆ hasSplittingSurface()
| bool regina::Triangulation< 3 >::hasSplittingSurface | ( | ) | const |
Determines whether this triangulation has a normal splitting surface.
See NormalSurface::isSplitting() for details regarding normal splitting surfaces.
In the special case where this is the empty triangulation, this routine returns false.
As of Regina 6.0, this routine is now fast (small polynomial time), and works even for triangulations with more than one connected component. Thanks to Robert Haraway.
- Returns
trueif and only if this triangulation has a normal splitting surface.
◆ hasStrictAngleStructure()
| bool regina::Triangulation< 3 >::hasStrictAngleStructure | ( | ) | const |
Determines whether this triangulation supports a strict angle structure.
Recall that a strict angle structure is one in which every angle is strictly between 0 and π.
This routine returns false if and only if strictAngleStructure() throws an exception. However, if you do not know whether a strict angle structure exists, then this routine is faster:
- If there is no strict angle structure, this routine will avoid the overhead of throwing and catching exceptions.
- If there is a strict angle structure, this routine will find and cache this angle structure, which means that any subsequent call to strictAngleStructure() to retrieve its details will be essentially instantaneous.
The underlying algorithm runs a single linear program (it does not enumerate all vertex angle structures). This means that it is likely to be fast even for large triangulations.
- Returns
trueif and only if a strict angle structure exists on this triangulation.
◆ hasTwoSphereBoundaryComponents()
|
inline |
Determines if this triangulation contains any two-sphere boundary components.
- Returns
trueif and only if there is at least one two-sphere boundary component.
◆ homology() [1/2]
|
inherited |
Returns the kth homology group of this triangulation, treating any ideal vertices as though they had been truncated.
A problem here is that, if dim is not one of Regina's standard dimensions, then Regina cannot actually detect ideal vertices (since in general this requires solving undecidable problems). Currently we resolve this by insisting that, in higher dimensions, the homology dimension k is at most (dim-2); the underlying algorithm will then effectively truncate all vertices (since truncating "ordinary" vertices whose links are spheres or balls does not affect the kth homology in such cases).
In general, this routine insists on working with a valid triangulation (see isValid() for what this means). However, for historical reasons, if you are computing first homology (k = 1) then your triangulation is allowed to be invalid, though the results might or might not be useful to you. The homology will be computed using the dual skeleton: what this means is that any invalid faces of dimension 0,1,...,(dim-3) will be treated as though their centroids had been truncated, but any invalid (dim-2)-faces will be treated without such truncation. A side-effect is that, after performing a barycentric on an invalid triangulation, the group returned by homology<1>() might change.
- Warning
- In dimension 3, if you are calling this from the subclass SnapPeaTriangulation then any fillings on the cusps will be ignored. (This is the same as for every routine implemented by Regina's Triangulation<3> class.) If you wish to compute homology with fillings, call SnapPeaTriangulation::homologyFilled() instead.
- Precondition
- Unless you are computing first homology (k = 1), this triangulation must be valid, and every face that is not a vertex must have a ball or sphere link. The link condition already forms part of the validity test if dim is one of Regina's standard dimensions, but in higher dimensions it is the user's own responsibility to ensure this. See isValid() for details.
- Exceptions
-
FailedPrecondition This triangulation is invalid, and the homology dimension k is not 1.
- Python
- Not present, since Python does not support templates. Python users can instead use the variant
homology(k).
- Template Parameters
-
k the dimension of the homology group to return; this must be between 1 and (dim - 1) inclusive if dim is one of Regina's standard dimensions, or between 1 and (dim - 2) inclusive if not.
- Returns
- the kth homology group.
◆ homology() [2/2]
|
inlineinherited |
Returns the kth homology group of this triangulation, treating any ideal vertices as though they had been truncated, where the parameter k does not need to be known until runtime.
For C++ programmers who know k at compile time, you are better off using the template function homology<k>() instead, which is slightly faster.
See the templated homology<k>() for full details on exactly what this function computes.
- Precondition
- Unless you are computing first homology (k = 1), this triangulation must be valid, and every face that is not a vertex must have a ball or sphere link. The link condition already forms part of the validity test if dim is one of Regina's standard dimensions, but in higher dimensions it is the user's own responsibility to ensure this. See isValid() for details.
- Exceptions
-
FailedPrecondition This triangulation is invalid, and the homology dimension k is not 1. InvalidArgument the homology dimension k is outside the supported range. This range depends upon the triangulation dimension dim; for details see the documentation below for the argument k.
- Python
- Like the C++ template function homology<k>(), you can omit the homology dimension k; this will default to 1.
- Parameters
-
k the dimension of the homology group to return; this must be between 1 and (dim - 1) inclusive if dim is one of Regina's standard dimensions, or between 1 and (dim - 2) inclusive if not.
- Returns
- the kth homology group.
◆ homologyBdry()
| const AbelianGroup & regina::Triangulation< 3 >::homologyBdry | ( | ) | const |
Returns the first homology group of the boundary for this triangulation.
Note that ideal vertices are considered part of the boundary.
Bear in mind that each time the triangulation changes, the homology groups will be deleted. Thus the reference that is returned from this routine should not be kept for later use. Instead, homologyBdry() should be called again; this will be instantaneous if the group has already been calculated.
This routine is fairly fast, since it deduces the homology of each boundary component through knowing what kind of surface it is.
- Precondition
- This triangulation is valid.
- Exceptions
-
FailedPrecondition This triangulation is invalid.
- Returns
- the first homology group of the boundary.
◆ homologyH1()
|
inlineinherited |
Deprecated routine that returns the first homology group for this triangulation.
- Deprecated:
- This is identical to calling homology<1>(), or just homology().
- Returns
- the first homology group.
◆ homologyH2()
|
inline |
Deprecated routine that returns the second homology group for this triangulation.
- Deprecated:
- This is identical to calling homology<2>().
- Precondition
- This triangulation is valid.
- Exceptions
-
FailedPrecondition This triangulation is invalid.
- Returns
- the second homology group.
◆ homologyH2Z2()
|
inline |
Returns the second homology group with coefficients in Z_2 for this triangulation.
If this triangulation contains any ideal vertices, the homology group will be calculated as if each such vertex had been truncated. The algorithm used calculates the relative first homology group with respect to the boundary and uses homology and cohomology theorems to deduce the second homology group.
This group will simply be the direct sum of several copies of Z_2, so the number of Z_2 terms is returned.
- Precondition
- This triangulation is valid.
- Exceptions
-
FailedPrecondition This triangulation is invalid.
- Returns
- the number of Z_2 terms in the second homology group with coefficients in Z_2.
◆ homologyRel()
| const AbelianGroup & regina::Triangulation< 3 >::homologyRel | ( | ) | const |
Returns the relative first homology group with respect to the boundary for this triangulation.
Note that ideal vertices are considered part of the boundary.
Bear in mind that each time the triangulation changes, the homology groups will be deleted. Thus the reference that is returned from this routine should not be kept for later use. Instead, homologyRel() should be called again; this will be instantaneous if the group has already been calculated.
- Precondition
- This triangulation is valid.
- Exceptions
-
FailedPrecondition This triangulation is invalid.
- Returns
- the relative first homology group with respect to the boundary.
◆ idealToFinite()
| bool regina::Triangulation< 3 >::idealToFinite | ( | ) |
Converts an ideal triangulation into a finite triangulation.
All ideal or invalid vertices are truncated and thus converted into real boundary components made from unglued faces of tetrahedra.
Note that this operation is a loose converse of finiteToIdeal().
- Warning
- Currently, this routine subdivides all tetrahedra as if all vertices (not just some) were ideal. This may lead to more tetrahedra than are necessary.
- Currently, the presence of an invalid edge will force the triangulation to be subdivided regardless of the value of parameter forceDivision. The final triangulation will still have the projective plane cusp caused by the invalid edge.
- Todo:
- Optimise (long-term): Have this routine only use as many tetrahedra as are necessary, leaving finite vertices alone.
- Returns
trueif and only if the triangulation was changed.
◆ inAnyPacket() [1/2]
| std::shared_ptr< Packet > regina::Triangulation< 3 >::inAnyPacket | ( | ) |
Returns the packet that holds this data, even if it is held indirectly via a SnapPea triangulation.
This routine is similar to PacketOf<Triangulation<3>>::packet(). In particular, if this triangulation is held directly by a 3-dimensional triangulation packet p, then this routine will return p.
The difference is when this triangulation is held "indirectly" by a SnapPea triangulation packet q (i.e., this is the inherited Triangulation<3> data belonging to the SnapPea triangulation). In such a scenario, Triangulation<3>::packet() will return null (since there is no "direct" 3-dimensional triangulation packet), but inAnyPacket() will return q (since the triangulation is still "indirectly" held by a different type of packet).
- Returns
- the packet that holds this data (directly or indirectly), or
nullif this data is not held by either a 3-dimensional triangulation packet or a SnapPea triangulation packet.
◆ inAnyPacket() [2/2]
| std::shared_ptr< const Packet > regina::Triangulation< 3 >::inAnyPacket | ( | ) | const |
Returns the packet that holds this data, even if it is held indirectly via a SnapPea triangulation.
This routine is similar to PacketOf<Triangulation<3>>::packet(). In particular, if this triangulation is held directly by a 3-dimensional triangulation packet p, then this routine will return p.
The difference is when this triangulation is held "indirectly" by a SnapPea triangulation packet q (i.e., this is the inherited Triangulation<3> data belonging to the SnapPea triangulation). In such a scenario, Triangulation<3>::packet() will return null (since there is no "direct" 3-dimensional triangulation packet), but inAnyPacket() will return q (since the triangulation is still "indirectly" held by a different type of packet).
- Returns
- the packet that holds this data (directly or indirectly), or
nullif this data is not held by either a 3-dimensional triangulation packet or a SnapPea triangulation packet.
◆ insertAugTriSolidTorus()
| void regina::Triangulation< 3 >::insertAugTriSolidTorus | ( | long | a1, |
| long | b1, | ||
| long | a2, | ||
| long | b2, | ||
| long | a3, | ||
| long | b3 | ||
| ) |
Deprecated routine that inserts an augmented triangular solid torus with the given parameters into this triangulation.
Almost all augmented triangular solid tori represent Seifert fibred spaces with three or fewer exceptional fibres. Augmented triangular solid tori are described in more detail in the AugTriSolidTorus class notes.
The resulting Seifert fibred space will be SFS((a1, b1), (a2, b2), (a3, b3), (1,1)), where the parameters a1, ..., b3 are passed as arguments to this routine. The three layered solid tori that are attached to the central triangular solid torus will be LST(|a1|, |b1|, |-a1-b1|), ..., LST(|a3|, |b3|, |-a3-b3|).
The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.
- Deprecated:
- If you just wish to create an augmented triangular solid torus, call
Example<3>::augTriSolidTorus(...). If you wish to insert a copy of it into an existing triangulation, callinsertTriangulation(Example<3>::augTriSolidTorus(...)).
- Precondition
- gcd(a1, b1) = 1.
- gcd(a2, b2) = 1.
- gcd(a3, b3) = 1.
- Parameters
-
a1 a parameter describing the first layered solid torus in the augmented triangular solid torus; this may be either positive or negative. b1 a parameter describing the first layered solid torus in the augmented triangular solid torus; this may be either positive or negative. a2 a parameter describing the second layered solid torus in the augmented triangular solid torus; this may be either positive or negative. b2 a parameter describing the second layered solid torus in the augmented triangular solid torus; this may be either positive or negative. a3 a parameter describing the third layered solid torus in the augmented triangular solid torus; this may be either positive or negative. b3 a parameter describing the third layered solid torus in the augmented triangular solid torus; this may be either positive or negative.
◆ insertConstruction()
|
inherited |
Deprecated routine that inserts a given triangulation into this triangulation, where the given triangulation is described by a pair of integer arrays.
The main purpose of this routine was to help users to hard-code triangulations into C++ source files. Nowadays you are better off doing this through fromGluings() instead.
This routine will insert an additional nSimplices top-dimensional simplices into this triangulation. We number these simplices 0,1,...,nSimplices-1. The gluings between these new simplices should be stored in the two arrays as follows.
The adjacencies array describes which simplices are joined to which others. Specifically, adjacencies[s][f] indicates which of the new simplices is joined to facet f of simplex s. This should be between 0 and nSimplices-1 inclusive, or -1 if facet f of simplex s is to be left as a boundary facet.
The gluings array describes the particular gluing permutations used to join these simplices together. Specifically, gluings[s][f][0..dim] should describe the permutation used to join facet f of simplex s to its adjacent simplex. These dim+1 integers should be 0,1,...,dim in some order, so that gluings[s][f][i] contains the image of i under this permutation. If facet f of simplex s is to be left as a boundary facet, then gluings[s][f][0..dim] may contain anything (and will be duly ignored).
If this triangulation is empty before this routine is called, then the new simplices will be given indices 0,1,...,nSimplices-1 according to the numbering described above. Otherwise they will be inserted after any pre-existing simplices, and so they will be given larger indices instead. In the latter case, the adjacencies array should still refer to the new simplices as 0,1,...,nSimplices-1, and this routine will handle any renumbering automatically at runtime.
It is the responsibility of the caller of this routine to ensure that the given arrays are correct and consistent. No error checking will be performed by this routine.
- Deprecated:
- Use the static routine fromGluings() instead. Code that uses fromGluings() is more compact and still readable; also fromGluings() performs error checking on the input.
- Python
- Not present; instead you can use fromGluings(), which is better in many ways (and which is not deprecated).
- Parameters
-
nSimplices the number of additional simplices to insert. adjacencies describes which simplices are adjace to which others, as described above. This array must have initial dimension at least nSimplices. gluings describes the specific gluing permutations, as described above. This array must also have initial dimension at least nSimplices.
◆ insertLayeredLensSpace()
| void regina::Triangulation< 3 >::insertLayeredLensSpace | ( | size_t | p, |
| size_t | q | ||
| ) |
Deprecated routine that inserts a new layered lens space L(p,q) into this triangulation.
The lens space will be created by gluing together two layered solid tori in a way that uses the fewest possible tetrahedra.
The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.
- Deprecated:
- If you just wish to create a layered lens space, call
Example<3>::lens(p, q). If you wish to insert a copy of it into an existing triangulation, callinsertTriangulation(Example<3>::lens(p, q)).
- Precondition
- p > q >= 0 unless (p,q) = (0,1);
- gcd(p, q) = 1.
- Parameters
-
p a parameter of the desired lens space. q a parameter of the desired lens space.
- See also
- LayeredLensSpace
◆ insertLayeredLoop()
| void regina::Triangulation< 3 >::insertLayeredLoop | ( | size_t | length, |
| bool | twisted | ||
| ) |
Deprecated routine that inserts a layered loop of the given length into this triangulation.
Layered loops are described in more detail in the LayeredLoop class notes.
The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.
- Parameters
-
length the length of the new layered loop; this must be strictly positive. twisted trueif the new layered loop should be twisted, orfalseif it should be untwisted.
- Deprecated:
- If you just wish to create a layered loop, call
Example<3>::layeredLoop(...). If you wish to insert a copy of it into an existing triangulation, callinsertTriangulation(Example<3>::layeredLoop(...)).
- See also
- LayeredLoop
◆ insertLayeredSolidTorus()
| Tetrahedron< 3 > * regina::Triangulation< 3 >::insertLayeredSolidTorus | ( | unsigned long | cuts0, |
| unsigned long | cuts1 | ||
| ) |
Inserts a new layered solid torus into the triangulation.
The meridinal disc of the layered solid torus will intersect the three edges of the boundary torus in cuts0, cuts1 and (cuts0 + cuts1) points respectively.
The boundary torus will always consist of faces 012 and 013 of the tetrahedron containing this boundary torus (this tetrahedron will be returned). In face 012, edges 12, 02 and 01 will meet the meridinal disc cuts0, cuts1 and (cuts0 + cuts1) times respectively. The only exceptions are if these three intersection numbers are (1,1,2) or (0,1,1), in which case edges 12, 02 and 01 will meet the meridinal disc (1, 2 and 1) or (1, 1 and 0) times respectively.
The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.
- Precondition
- 0 <= cuts0 <= cuts1;
- gcd(cuts0, cuts1) = 1.
- Parameters
-
cuts0 the smallest of the three desired intersection numbers. cuts1 the second smallest of the three desired intersection numbers.
- Returns
- the tetrahedron containing the boundary torus.
- See also
- LayeredSolidTorus
◆ insertRehydration()
|
inline |
Deprecated routine that inserts the rehydration of the given string into this triangulation.
See dehydrate() and rehydrate() for more information on dehydration strings.
This routine will first rehydrate the given string into a proper triangulation. The tetrahedra from the rehydrated triangulation will then be inserted into this triangulation in the same order in which they appear in the rehydrated triangulation, and the numbering of their vertices (0-3) will not change.
- Deprecated:
- If you just wish to rehydrate a dehydration string, simply call the static routine rehydrate(). If you wish to insert the rehydration into an existing triangulation, call rehydrate() followed by insertTriangulation().
- Exceptions
-
InvalidArgument the given string could not be rehydrated.
- Parameters
-
dehydration a dehydrated representation of the triangulation to insert. Case is irrelevant; all letters will be treated as if they were lower case.
◆ insertSFSOverSphere()
| void regina::Triangulation< 3 >::insertSFSOverSphere | ( | long | a1 = 1, |
| long | b1 = 0, |
||
| long | a2 = 1, |
||
| long | b2 = 0, |
||
| long | a3 = 1, |
||
| long | b3 = 0 |
||
| ) |
Deprecated routine that inserts an orientable Seifert fibred space with at most three exceptional fibres over the 2-sphere into this triangulation.
The Seifert fibred space will be SFS((a1, b1), (a2, b2), (a3, b3)), where the parameters a1, ..., b3 are passed as arguments to this routine.
The three pairs of parameters (a, b) do not need to be normalised, i.e., the parameters can be positive or negative and b may lie outside the range [0..a). There is no separate twisting parameter; each additional twist can be incorporated into the existing parameters by replacing some pair (a>, b) with the pair (a, a + b). For Seifert fibred spaces with less than three exceptional fibres, some or all of the parameter pairs may be (1, k) or even (1, 0).
The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.
- Deprecated:
- If you just wish to triangulate the given Seifert fibred space, call
Example<3>::sfsOverSphere(...). If you wish to insert a copy of it into an existing triangulation, callinsertTriangulation(Example<3>::sfsOverSphere(...)).
- Precondition
- None of a1, a2 or a3 are 0.
- gcd(a1, b1) = 1.
- gcd(a2, b2) = 1.
- gcd(a3, b3) = 1.
- Parameters
-
a1 a parameter describing the first exceptional fibre. b1 a parameter describing the first exceptional fibre. a2 a parameter describing the second exceptional fibre. b2 a parameter describing the second exceptional fibre. a3 a parameter describing the third exceptional fibre. b3 a parameter describing the third exceptional fibre.
◆ insertTriangulation()
|
inherited |
Inserts a copy of the given triangulation into this triangulation.
The top-dimensional simplices of source will be copied into this triangulation in the same order in which they appear in source. That is, if the original size of this triangulation was S, then the simplex at index i in source will be copied into this triangulation as a new simplex at index S+i.
The copies will use the same vertex numbering and descriptions as the original simplices from source, and any gluings between the simplices of source will likewise be copied across as gluings between their copies in this triangulation.
This routine behaves correctly when source is this triangulation.
- Parameters
-
source the triangulation whose copy will be inserted.
◆ intelligentSimplify()
| bool regina::Triangulation< 3 >::intelligentSimplify | ( | ) |
Attempts to simplify the triangulation using fast and greedy heuristics.
This routine will attempt to reduce both the number of tetrahedra and the number of boundary triangles (with the number of tetrahedra as its priority).
Currently this routine uses simplifyToLocalMinimum() in combination with random 4-4 moves, book opening moves and book closing moves.
Although intelligentSimplify() works very well most of the time, it can occasionally get stuck; in such cases you may wish to try the more powerful but (much) slower simplifyExhaustive() instead.
If this triangulation is currently oriented, then this operation will preserve the orientation.
- Warning
- Running this routine multiple times upon the same triangulation may return different results, since the implementation makes random decisions. More broadly, the implementation of this routine (and therefore its results) may change between different releases of Regina.
- Todo:
- Optimise: Include random 2-3 moves to get out of wells.
- Returns
trueif and only if the triangulation was successfully simplified. Otherwise this triangulation will not be changed.
◆ isBall()
| bool regina::Triangulation< 3 >::isBall | ( | ) | const |
Determines whether this is a triangulation of a 3-dimensional ball.
This routine is based on isSphere(), which in turn combines Rubinstein's 3-sphere recognition algorithm with Jaco and Rubinstein's 0-efficiency prime decomposition algorithm.
- Warning
- The algorithms used in this routine rely on normal surface theory and so can be very slow for larger triangulations (although faster tests are used where possible). The routine knowsBall() can be called to see if this property is already known or if it happens to be very fast to calculate for this triangulation.
- Returns
trueif and only if this is a triangulation of a 3-dimensional ball.
◆ isClosed()
|
inline |
Determines if this triangulation is closed.
This is the case if and only if it has no boundary. Note that ideal triangulations are not closed.
- Returns
trueif and only if this triangulation is closed.
◆ isConnected()
|
inlineinherited |
Determines if this triangulation is connected.
This routine returns false only if there is more than one connected component. In particular, it returns true for the empty triangulation.
- Returns
trueif and only if this triangulation is connected.
◆ isContainedIn()
|
inlineinherited |
Determines if an isomorphic copy of this triangulation is contained within the given triangulation, possibly as a subcomplex of some larger component (or components).
Specifically, this routine determines if there is a boundary incomplete combinatorial isomorphism from this triangulation to other. Boundary incomplete isomorphisms are described in detail in the Isomorphism class notes.
In particular, note that facets of top-dimensional simplices that lie on the boundary of this triangulation need not correspond to boundary facets of other, and that other may contain more top-dimensional simplices than this triangulation.
If a boundary incomplete isomorphism is found, the details of this isomorphism are returned. Thus, to test whether an isomorphism exists, you can just call if (isContainedIn(other)).
If more than one such isomorphism exists, only one will be returned. For a routine that returns all such isomorphisms, see findAllSubcomplexesIn().
- Warning
- For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
- Parameters
-
other the triangulation in which to search for an isomorphic copy of this triangulation.
- Returns
- details of the isomorphism if such a copy is found, or no value otherwise.
◆ isEmpty()
|
inlineinherited |
Determines whether this triangulation is empty.
An empty triangulation is one with no simplices at all.
- Returns
trueif and only if this triangulation is empty.
◆ isHaken()
| bool regina::Triangulation< 3 >::isHaken | ( | ) | const |
Determines whether the underlying 3-manifold (which must be closed and orientable) is Haken.
In other words, this routine determines whether the underlying 3-manifold contains an embedded closed two-sided incompressible surface.
Currently Hakenness testing is available only for irreducible manifolds. This routine will first test whether the manifold is irreducible and, if it is not, will return false immediately.
- Precondition
- This triangulation is valid, closed, orientable and connected.
- Warning
- This routine could be very slow for larger triangulations.
- Returns
trueif and only if the underlying 3-manifold is irreducible and Haken.
◆ isIdeal()
|
inline |
Determines if this triangulation is ideal.
This is the case if and only if one of the vertex links is closed and not a 2-sphere. Note that the triangulation is not required to be valid.
- Returns
trueif and only if this triangulation is ideal.
◆ isIdenticalTo()
|
inlineinherited |
Deprecated routine that determines if this triangulation is combinatorially identical to the given triangulation.
- Deprecated:
- This routine has been renamed to the comparison operator (==).
- Parameters
-
other the triangulation to compare with this.
- Returns
trueif and only if the two triangulations are combinatorially identical.
◆ isIrreducible()
| bool regina::Triangulation< 3 >::isIrreducible | ( | ) | const |
Determines whether the underlying 3-manifold (which must be closed) is irreducible.
In other words, this routine determines whether every embedded sphere in the underlying 3-manifold bounds a ball.
If the underlying 3-manifold is orientable, this routine will use fast crushing and branch-and-bound methods. If the underlying 3-manifold is non-orientable, it will use a (much slower) full enumeration of vertex normal surfaces.
- Warning
- The algorithms used in this routine rely on normal surface theory and might be slow for larger triangulations.
- Precondition
- This triangulation is valid, closed, orientable and connected.
- Returns
trueif and only if the underlying 3-manifold is irreducible.
◆ isIsomorphicTo()
|
inlineinherited |
Determines if this triangulation is combinatorially isomorphic to the given triangulation.
Two triangulations are isomorphic if and only it is possible to relabel their top-dimensional simplices and the (dim+1) vertices of each simplex in a way that makes the two triangulations combinatorially identical, as returned by isIdenticalTo().
Equivalently, two triangulations are isomorphic if and only if there is a one-to-one and onto boundary complete combinatorial isomorphism from this triangulation to other, as described in the Isomorphism class notes.
In particular, note that this triangulation and other must contain the same number of top-dimensional simplices for such an isomorphism to exist.
If the triangulations are isomorphic, then this routine returns one such boundary complete isomorphism (i.e., one such relabelling). Otherwise it returns no value. Thus, to test whether an isomorphism exists, you can just call if (isIsomorphicTo(other)).
There may be many such isomorphisms between the two triangulations. If you need to find all such isomorphisms, you may call findAllIsomorphisms() instead.
If you need to ensure that top-dimensional simplices are labelled the same in both triangulations (i.e., that the triangulations are related by the identity isomorphism), you should call the stricter test isIdenticalTo() instead.
- Warning
- For large dimensions, this routine can become extremely slow: its running time includes a factor of (dim+1)!.
- Todo:
- Optimise: Improve the complexity by choosing a simplex mapping from each component and following gluings to determine the others.
- Parameters
-
other the triangulation to compare with this one.
- Returns
- details of the isomorphism if the two triangulations are combinatorially isomorphic, or no value otherwise.
◆ isOrdered()
| bool regina::Triangulation< 3 >::isOrdered | ( | ) | const |
Determines if this triangulation is ordered; that is, if tetrahedron vertices are labelled so that all gluing permutations are order-preserving on the tetrahedron faces.
Equivalently, this tests whether the edges of the triangulation can all be oriented such that they induce a consistent ordering on the vertices of each tetrahedron.
Triangulations are not ordered by default, and indeed some cannot be ordered at all. The routine order() will attempt to relabel tetrahedron vertices to give an ordered triangulation.
- Returns
trueif and only if all gluing permutations are order preserving on the tetrahedron faces.
◆ isOrientable()
|
inlineinherited |
Determines if this triangulation is orientable.
- Returns
trueif and only if this triangulation is orientable.
◆ isOriented()
|
inherited |
Determines if this triangulation is oriented; that is, if the vertices of its top-dimensional simplices are labelled in a way that preserves orientation across adjacent facets.
Specifically, this routine returns true if and only if every gluing permutation has negative sign.
Note that orientable triangulations are not always oriented by default. You can call orient() if you need the top-dimensional simplices to be oriented consistently as described above.
A non-orientable triangulation can never be oriented.
- Returns
trueif and only if all top-dimensional simplices are oriented consistently.
◆ isoSig()
|
inherited |
Constructs the isomorphism signature of the given type for this triangulation.
Support for different types of signature is new to Regina 7.0 (see below for details); all isomorphism signatures created in Regina 6.0.1 or earlier are of the default type IsoSigClassic.
An isomorphism signature is a compact representation of a triangulation that uniquely determines the triangulation up to combinatorial isomorphism. That is, for any fixed signature type T, two triangulations of dimension dim are combinatorially isomorphic if and only if their isomorphism signatures of type T are the same.
The length of an isomorphism signature is proportional to n log n, where n is the number of top-dimenisonal simplices. The time required to construct it is worst-case O((dim!) n^2 log^2 n). Whilst this is fine for large triangulations, it becomes very slow for large dimensions; the main reason for introducing different signature types is that some alternative types can be much faster to compute in practice.
Whilst the format of an isomorphism signature bears some similarity to dehydration strings for 3-manifolds, they are more general: isomorphism signatures can be used with any triangulations, including closed, bounded and/or disconnected triangulations, as well as triangulations with many simplices. Note also that 3-manifold dehydration strings are not unique up to isomorphism (they depend on the particular labelling of tetrahedra).
The routine fromIsoSig() can be used to recover a triangulation from an isomorphism signature (only if the default encoding has been used, but it does not matter which signature type was used). The triangulation recovered might not be identical to the original, but it will be combinatorially isomorphic. If you need the precise relabelling, you can call isoSigDetail() instead.
Regina supports several different variants of isomorphism signatures, which are tailored to different computational needs; these are currently determined by the template parameters Type and Encoding:
- The Type parameter identifies which signature type is to be constructed. Essentially, different signature types use different rules to determine which labelling of a triangulation is "canonical". The default type IsoSigClassic is slow (it never does better than the worst-case time described above); its main advantage is that it is consistent with the original implementation of isomorphism signatures in Regina 4.90.
- The Encoding parameter controls how Regina encodes a canonical labelling into a final signature. The default encoding IsoSigPrintable returns a std::string consisting entirely of printable characters in the 7-bit ASCII range. Importantly, this default encoding is currently the only encoding from which Regina can reconstruct a triangulation from its isomorphism signature.
You may instead pass your own type and/or encoding parameters as template arguments. Currently this facility is for internal use only, and the requirements for type and encoding parameters may change in future versions of Regina. At present:
- The Type parameter should be a class that is constructible from a componenent reference, and that offers the member functions simplex(), perm() and next(); see the implementation of IsoSigClassic for details.
- The Encoding parameter should be a class that offers a Signature type alias, and static functions emptySig() and encode(). See the implementation of IsoSigPrintable for details.
For a full and precise description of the classic isomorphism signature format for 3-manifold triangulations, see Simplification paths in the Pachner graphs of closed orientable 3-manifold triangulations, Burton, 2011, arXiv:1110.6080. The format for other dimensions is essentially the same, but with minor dimension-specific adjustments.
- Python
- Although this is a templated function, all of the variants supplied with Regina are available to Python users. For the default type and encoding, you can simply call isoSig(). For other signature types, you should call the function as isoSig_Type, where the suffix Type is an abbreviated version of the Type template parameter; one such example is
isoSig_EdgeDegrees(for the case where Type is the class IsoSigEdgeDegrees). Currently Regina only offers one encoding (the default), and so there are no suffixes for encodings.
- Warning
- Do not mix isomorphism signatures between dimensions! It is possible that the same string could corresponding to both a p-dimensional triangulation and a q-dimensional triangulation for different dimensions p and q.
- Returns
- the isomorphism signature of this triangulation.
◆ isoSigComponentSize()
|
staticinherited |
Deduces the number of top-dimensional simplices in a connected triangulation from its isomorphism signature.
See isoSig() for more information on isomorphism signatures. It will be assumed that the signature describes a triangulation of dimension dim.
Currently this routine only supports isomorphism signatures that were created with the default encoding (i.e., there was no Encoding template parameter passed to isoSig()).
If the signature describes a connected triangulation, this routine will simply return the size of that triangulation (e.g., the number of tetrahedra in the case dim = 3). You can also pass an isomorphism signature that describes a disconnected triangulation; however, this routine will only return the number of top-dimensional simplices in the first connected component. If you need the total size of a disconnected triangulation, you will need to reconstruct the full triangulation by calling fromIsoSig() instead.
This routine is very fast, since it only examines the first few characters of the isomorphism signature (in which the size of the first component is encoded). However, a side-effect of this is that it is possible to pass an invalid isomorphism signature and still receive a positive result. If you need to test whether a signature is valid or not, you must call fromIsoSig() instead, which will examine the entire signature in full.
- Warning
- Do not mix isomorphism signatures between dimensions! It is possible that the same string could corresponding to both a p-dimensional triangulation and a q-dimensional triangulation for different dimensions p and q.
- Parameters
-
sig the isomorphism signature of a dim-dimensional triangulation. Note that isomorphism signature are case-sensitive (unlike, for example, dehydration strings for 3-manifolds).
- Returns
- the number of top-dimensional simplices in the first connected component, or 0 if this could not be determined because the given string was not a valid isomorphism signature created using the default encoding.
◆ isoSigDetail()
|
inherited |
Constructs the isomorphism signature for this triangulation, along with the relabelling that will occur when the triangulation is reconstructed from it.
Essentially, an isomorphism signature is a compact representation of a triangulation that uniquely determines the triangulation up to combinatorial isomorphism. See isoSig() for much more detail on isomorphism signatures as well as the support for different signature types and encodings.
As described in the isoSig() notes, you can call fromIsoSig() to recover a triangulation from an isomorphism signature (assuming the default encoding was used). Whilst the triangulation that is recovered will be combinatorially isomorphic to the original, it might not be identical. This routine returns not only the isomorphism signature, but also an isomorphism that describes the precise relationship between this triangulation and the reconstruction from fromIsoSig().
Specifically, if this routine returns the pair (sig, relabelling), this means that the triangulation reconstructed from fromIsoSig(sig) will be identical to relabelling.apply(this).
- Python
- Although this is a templated function, all of the variants supplied with Regina are available to Python users. For the default type and encoding, you can simply call isoSigDetail(). For other signature types, you should call the function as isoSigDetail_Type, where the suffix Type is an abbreviated version of the Type template parameter; one such example is
isoSigDetail_EdgeDegrees(for the case where Type is the class IsoSigEdgeDegrees). Currently Regina only offers one encoding (the default), and so there are no suffixes for encodings.
- Precondition
- This triangulation must be non-empty and connected. The facility to return a relabelling for disconnected triangulations may be added to Regina in a later release.
- Exceptions
-
FailedPrecondition This triangulation is either empty or disconnected.
- Returns
- a pair containing (i) the isomorphism signature of this triangulation, and (ii) the isomorphism between this triangulation and the triangulation that would be reconstructed from fromIsoSig().
◆ isReadOnlySnapshot()
|
inlineprotectedinherited |
Determines if this object is a read-only deep copy that was created by a snapshot.
Recall that, if an image I of type T has a snapshot pointing to it, and if that image I is about to be modified or destroyed, then the snapshot will make an internal deep copy of I and refer to that instead.
The purpose of this routine is to identify whether the current object is such a deep copy. This may be important information, since a snapshot's deep copy is read-only: it must not be modified or destroyed by the outside world. (Of course the only way to access this deep copy is via const reference from the SnapshotRef dereference operators, but there are settings in which this constness is "forgotten", such as Regina's Python bindings.)
- Python
- Although this Snapshottable base class is not visible to Python, the isReadOnlySnapshot() function is still visible as a member function of the class T.
- Returns
trueif and only if this object is a deep copy that was taken by a Snapshot object of some original type T image.
◆ isSnapPea() [1/2]
| SnapPeaTriangulation * regina::Triangulation< 3 >::isSnapPea | ( | ) |
Returns the SnapPea triangulation that holds this data, if there is one.
This routine essentially replaces a dynamic_cast, since the class Triangulation<3> is not polymorphic.
If this object in fact belongs to a SnapPeaTriangulation t (through its inherited Triangulation<3> interface), then this routine will return t. Otherwise it will return null.
- Returns
- the SnapPea triangulation that holds this data, or
nullif this data is not part of a SnapPea triangulation.
◆ isSnapPea() [2/2]
| const SnapPeaTriangulation * regina::Triangulation< 3 >::isSnapPea | ( | ) | const |
Returns the SnapPea triangulation that holds this data, if there is one.
This routine essentially replaces a dynamic_cast, since the class Triangulation<3> is not polymorphic.
If this object in fact belongs to a SnapPeaTriangulation t (through its inherited Triangulation<3> interface), then this routine will return t. Otherwise it will return null.
- Returns
- the SnapPea triangulation that holds this data, or
nullif this data is not part of a SnapPea triangulation.
◆ isSolidTorus()
| bool regina::Triangulation< 3 >::isSolidTorus | ( | ) | const |
Determines whether this is a triangulation of the solid torus; that is, the unknot complement.
This routine can be used on a triangulation with real boundary triangles, or on an ideal triangulation (in which case all ideal vertices will be assumed to be truncated).
- Warning
- The algorithms used in this routine rely on normal surface theory and so might be very slow for larger triangulations (although faster tests are used where possible). The routine knowsSolidTorus() can be called to see if this property is already known or if it happens to be very fast to calculate for this triangulation.
- Returns
trueif and only if this is either a real (compact) or ideal (non-compact) triangulation of the solid torus.
◆ isSphere()
| bool regina::Triangulation< 3 >::isSphere | ( | ) | const |
Determines whether this is a triangulation of a 3-sphere.
This routine relies upon a combination of Rubinstein's 3-sphere recognition algorithm and Jaco and Rubinstein's 0-efficiency prime decomposition algorithm.
- Warning
- The algorithms used in this routine rely on normal surface theory and so can be very slow for larger triangulations (although faster tests are used where possible). The routine knowsSphere() can be called to see if this property is already known or if it happens to be very fast to calculate for this triangulation.
- Returns
trueif and only if this is a 3-sphere triangulation.
◆ isStandard()
|
inline |
Determines if this triangulation is standard.
This is the case if and only if every vertex is standard. See Vertex<3>::isStandard() for further details.
- Returns
trueif and only if this triangulation is standard.
◆ isThreeSphere()
|
inline |
Deprecated function to test if this is a 3-sphere triangulation.
- Deprecated:
- This routine has been renamed isSphere().
- Returns
trueif and only if this is the 3-sphere.
◆ isTxI()
| bool regina::Triangulation< 3 >::isTxI | ( | ) | const |
Determines whether or not the underlying 3-manifold is the product of a torus with an interval.
This routine can be used on a triangulation with real boundary triangles, or ideal boundary components, or a mix of both. If there are any ideal vertices, they will be treated as though they were truncated.
The underlying algorithm is due to Robert C. Haraway, III; see https://journals.carleton.ca/jocg/index.php/jocg/article/view/433 for details.
- Warning
- This algorithm ultimately relies on isSolidTorus(), which might run slowly for large triangulations.
- Returns
trueif and only if this is a triangulation (either real, ideal or a combination) of the product of the torus with an interval.
◆ isValid()
|
inlineinherited |
Determines if this triangulation is valid.
There are several conditions that might make a dim-dimensional triangulation invalid:
- if some face is identified with itself under a non-identity permutation (e.g., an edge is identified with itself in reverse, or a triangle is identified with itself under a rotation);
- if some subdim-face does not have an appropriate link. Here the meaning of "appropriate" depends upon the type of face:
- for a face that belongs to some boundary facet(s) of this triangulation, its link must be a topological ball;
- for a vertex that does not belong to any boundary facets, its link must be a closed (dim - 1)-manifold;
- for a (subdim ≥ 1)-face that does not belong to any boundary facets, its link must be a topological sphere.
Condition (1) is tested for all dimensions dim. Condition (2) is more difficult, since it relies on undecidable problems. As a result, (2) is only tested when dim is one of Regina's standard dimensions.
If a triangulation is invalid then you can call Face<dim, subdim>::isValid() to discover exactly which face(s) are responsible, and you can call Face<dim, subdim>::hasBadIdentification() and/or Face<dim, subdim>::hasBadLink() to discover exactly which conditions fail.
Note that all invalid vertices are considered to be on the boundary; see isBoundary() for details.
- Returns
trueif and only if this triangulation is valid.
◆ isZeroEfficient()
| bool regina::Triangulation< 3 >::isZeroEfficient | ( | ) | const |
Determines if this triangulation is 0-efficient.
A triangulation is 0-efficient if its only normal spheres and discs are vertex linking, and if it has no 2-sphere boundary components.
- Returns
trueif and only if this triangulation is 0-efficient.
◆ knowsBall()
| bool regina::Triangulation< 3 >::knowsBall | ( | ) | const |
Is it already known (or trivial to determine) whether or not this is a triangulation of a 3-dimensional ball? See isBall() for further details.
If this property is indeed already known, future calls to isBall() will be very fast (simply returning the precalculated value).
If this property is not already known, this routine will nevertheless run some very fast preliminary tests to see if the answer is obviously no. If so, it will store false as the precalculated value for isBall() and this routine will return true.
Otherwise a call to isBall() may potentially require more significant work, and so this routine will return false.
- Warning
- This routine does not actually tell you whether this triangulation forms a ball; it merely tells you whether the answer has already been computed (or is very easily computed).
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsCompressingDisc()
| bool regina::Triangulation< 3 >::knowsCompressingDisc | ( | ) | const |
Is it already known (or trivial to determine) whether or not the underlying 3-manifold contains a compressing disc? See hasCompressingDisc() for further details.
If this property is indeed already known, future calls to hasCompressingDisc() will be very fast (simply returning the precalculated value).
If this property is not already known, this routine will nevertheless run some very fast preliminary tests to see if the answer is obviously no. If so, it will store false as the precalculated value for hasCompressingDisc() and this routine will return true.
Otherwise a call to hasCompressingDisc() may potentially require more significant work, and so this routine will return false.
- Warning
- This routine does not actually tell you whether the underlying 3-manifold has a compressing disc; it merely tells you whether the answer has already been computed (or is very easily computed).
- Precondition
- This triangulation is valid and is not ideal.
- The underlying 3-manifold is irreducible.
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsHaken()
| bool regina::Triangulation< 3 >::knowsHaken | ( | ) | const |
Is it already known (or trivial to determine) whether or not the underlying 3-manifold is Haken? See isHaken() for further details.
If this property is indeed already known, future calls to isHaken() will be very fast (simply returning the precalculated value).
- Warning
- This routine does not actually tell you whether the underlying 3-manifold is Haken; it merely tells you whether the answer has already been computed (or is very easily computed).
- Precondition
- This triangulation is valid, closed, orientable and connected.
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsIrreducible()
| bool regina::Triangulation< 3 >::knowsIrreducible | ( | ) | const |
Is it already known (or trivial to determine) whether or not the underlying 3-manifold is irreducible? See isIrreducible() for further details.
If this property is indeed already known, future calls to isIrreducible() will be very fast (simply returning the precalculated value).
- Warning
- This routine does not actually tell you whether the underlying 3-manifold is irreducible; it merely tells you whether the answer has already been computed (or is very easily computed).
- Precondition
- This triangulation is valid, closed, orientable and connected.
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsSolidTorus()
| bool regina::Triangulation< 3 >::knowsSolidTorus | ( | ) | const |
Is it already known (or trivial to determine) whether or not this is a triangulation of a solid torus (that is, the unknot complement)? See isSolidTorus() for further details.
If this property is indeed already known, future calls to isSolidTorus() will be very fast (simply returning the precalculated value).
If this property is not already known, this routine will nevertheless run some very fast preliminary tests to see if the answer is obviously no. If so, it will store false as the precalculated value for isSolidTorus() and this routine will return true.
Otherwise a call to isSolidTorus() may potentially require more significant work, and so this routine will return false.
- Warning
- This routine does not actually tell you whether this triangulation forms a solid torus; it merely tells you whether the answer has already been computed (or is very easily computed).
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsSphere()
| bool regina::Triangulation< 3 >::knowsSphere | ( | ) | const |
Is it already known (or trivial to determine) whether or not this is a triangulation of a 3-sphere? See isSphere() for further details.
If this property is indeed already known, future calls to isSphere() will be very fast (simply returning the precalculated value).
If this property is not already known, this routine will nevertheless run some very fast preliminary tests to see if the answer is obviously no. If so, it will store false as the precalculated value for isSphere() and this routine will return true.
Otherwise a call to isSphere() may potentially require more significant work, and so this routine will return false.
- Warning
- This routine does not actually tell you whether this triangulation forms a 3-sphere; it merely tells you whether the answer has already been computed (or is very easily computed).
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsStrictAngleStructure()
| bool regina::Triangulation< 3 >::knowsStrictAngleStructure | ( | ) | const |
Is it already known (or trivial to determine) whether or not this triangulation supports a strict angle structure? See hasStrictAngleStructure() for further details.
If this property is indeed already known, future calls to strictAngleStructure() and hasStrictAngleStructure() will be very fast (simply returning the precalculated solution).
- Warning
- This routine does not actually tell you whether the triangulation supports a strict angle structure; it merely tells you whether the answer has already been computed (or is very easily computed).
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsThreeSphere()
|
inline |
Deprecated function to determine if it is already known (or trivial to determine) whether this is a 3-sphere triangulation.
- Deprecated:
- This routine has been renamed knowsSphere().
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsTxI()
| bool regina::Triangulation< 3 >::knowsTxI | ( | ) | const |
Is it already known (or trivial to determine) whether or not this is a triangulation of a the product of a torus with an interval? See isTxI() for further details.
If this property is indeed already known, future calls to isTxI() will be very fast (simply returning the precalculated value).
If this property is not already known, this routine will nevertheless run some very fast preliminary tests to see if the answer is obviously no. If so, it will store false as the precalculated value for isTxI() and this routine will return true.
Otherwise a call to isTxI() may potentially require more significant work, and so this routine will return false.
- Warning
- This routine does not actually tell you whether this triangulation forms the product of the torus with an interval; it merely tells you whether the answer has already been computed (or is very easily computed).
- Returns
trueif and only if this property is already known or trivial to calculate.
◆ knowsZeroEfficient()
|
inline |
Is it already known whether or not this triangulation is 0-efficient? See isZeroEfficient() for further details.
If this property is already known, future calls to isZeroEfficient() will be very fast (simply returning the precalculated value).
- Warning
- This routine does not actually tell you whether this triangulation is 0-efficient; it merely tells you whether the answer has already been computed.
- Returns
trueif and only if this property is already known.
◆ layerOn()
| Tetrahedron< 3 > * regina::Triangulation< 3 >::layerOn | ( | Edge< 3 > * | edge | ) |
Performs a layering upon the given boundary edge of the triangulation.
See the Layering class notes for further details on what a layering entails.
The new tetrahedron will be returned, and the new boundary edge that it creates will be edge 5 (i.e., the edge joining vertices 2 and 3) of this tetrahedron.
- Precondition
- The given edge is a boundary edge of this triangulation, and the two boundary triangles on either side of it are distinct.
- Parameters
-
edge the boundary edge upon which to layer.
- Returns
- the new tetrahedron provided by the layering.
◆ longitude()
| Edge< 3 > * regina::Triangulation< 3 >::longitude | ( | ) |
Modifies a triangulated knot complement so that the algebraic longitude follows a single boundary edge, and returns this edge.
Assuming that this triangulation represents the complement of a knot in the 3-sphere, this routine:
- identifies the algebraic longitude of the knot complement; that is, identifies the non-trivial simple closed curve on the boundary whose homology in the 3-manifold is trivial;
- layers additional tetrahedra on the boundary if necessary so that this curve is represented by a single boundary edge;
- returns that (possibly new) boundary edge.
Whilst this routine returns less information than meridianLongitude(), it (1) runs much faster since it is based on fast algebraic calculations, and (2) guarantees to terminate. In contrast, meridianLongitude() must repeatedly try to test for 3-spheres, and (as a result of only using fast 3-sphere recognition heuristics) does not guarantee to terminate.
At present this routine is fairly restrictive in what triangulations it can work with: it requires the triangulation to be one-vertex and have real (not ideal) boundary. These restrictions may be eased in future versions of Regina.
If the algebraic longitude is already represented by a single boundary edge, then it is guaranteed that this routine will not modify the triangulation, and will simply return this boundary edge.
- Precondition
- The underlying 3-manifold is known to be the complement of a knot in the 3-sphere.
- This triangulation has precisely one vertex, and its (unique) boundary component is formed from two triangles.
- Warning
- This routine may modify the triangluation, as explained above, which will have the side-effect of invalidating any existing Vertex, Edge or Triangle references.
- If you have an ideal triangulation of a knot complement, you must first run idealToFinite() and then simplify the resulting triangulation to have two boundary triangles.
- Returns
- the boundary edge representing the algebraic longitude of the knot (after this triangulation has been modified if necessary), or
nullif an error (such as an integer overflow) occurred during the computation.
◆ makeCanonical()
|
inherited |
Relabel the top-dimensional simplices and their vertices so that this triangulation is in canonical form.
This is essentially the lexicographically smallest labelling when the facet gluings are written out in order.
Two triangulations are isomorphic if and only if their canonical forms are identical.
The lexicographic ordering assumes that the facet gluings are written in order of simplex index and then facet number. Each gluing is written as the destination simplex index followed by the gluing permutation (which in turn is written as the images of 0,1,...,dim in order).
- Precondition
- This routine currently works only when the triangulation is connected. It may be extended to work with disconnected triangulations in later versions of Regina.
- Returns
trueif the triangulation was changed, orfalseif the triangulation was in canonical form to begin with.
◆ makeDoubleCover()
|
inherited |
Converts this triangulation into its double cover.
Each orientable component will be duplicated, and each non-orientable component will be converted into its orientable double cover.
◆ markedHomology() [1/2]
|
inlineinherited |
Returns the kth homology group of this triangulation, without truncating ideal vertices, but with explicit coordinates that track the individual k-faces of this triangulation.
This is a specialised homology routine; you should only use it if you need to understand how individual k-faces (or chains of k-faces) appear within the homology group.
- The major disadvantage of this routine is that it does not truncate ideal vertices. Instead it computes the homology of the union of all top-dimensional simplices, working directly with the boundary maps between (k+1)-faces, k-faces and (k-1)-faces of the triangulation. If your triangulation is ideal, then this routine will almost certainly not give the correct homology group for the underlying manifold. If, however, all of your vertex links are spheres or balls (i.e., the triangulation is closed or all of its boundary components are built from unglued (dim-1)-faces), then the homology of the manifold will be computed correctly.
- The major advantage is that, instead of returning a simpler AbelianGroup, this routine returns a MarkedAbelianGroup. This allows you to track chains of individual k-faces of the triangulation as they appear within the homology group. Specifically, the chain complex cordinates with this MarkedAbelianGroup represent precisely the k-faces of the triangulation in the same order as they appear in the list faces<k>(), using the inherent orientation provided by Face<dim, k>.
- Precondition
- This triangulation is valid and non-empty.
- Exceptions
-
FailedPrecondition This triangulation is empty or invalid.
- Python
- Not present, since Python does not support templates. Python users can instead use the variant
markedHomology(k).
- Template Parameters
-
k the dimension of the homology group to compute; this must be between 1 and (dim-1) inclusive.
- Returns
- the kth homology group of the union of all simplices in this triangulation, as described above.
◆ markedHomology() [2/2]
|
inlineinherited |
Returns the kth homology group of this triangulation, without truncating ideal vertices, but with explicit coordinates that track the individual k-faces of this triangulation, where the parameter k does not need to be known until runtime.
For C++ programmers who know k at compile time, you are better off using the template function markedHomology<k>() instead, which is slightly faster.
See the templated markedHomology<k>() for full details on what this function computes, some important caveats to be aware of, and how the group that it returns should be interpreted.
- Precondition
- This triangulation is valid and non-empty.
- Exceptions
-
FailedPrecondition This triangulation is empty or invalid. InvalidArgument the homology dimension k is outside the supported range (i.e., less than 1 or greater than or equal to dim).
- Python
- Like the C++ template function markedHomology<k>(), you can omit the homology dimension k; this will default to 1.
- Parameters
-
k the dimension of the homology group to compute; this must be between 1 and (dim-1) inclusive.
- Returns
- the kth homology group of the union of all simplices in this triangulation, as described above.
◆ maximalForestInBoundary()
| std::set< Edge< 3 > * > regina::Triangulation< 3 >::maximalForestInBoundary | ( | ) | const |
Produces a maximal forest in the 1-skeleton of the triangulation boundary.
Note that the edge pointers returned will become invalid once the triangulation has changed.
- Returns
- a set containing the edges of the maximal forest.
◆ maximalForestInSkeleton()
| std::set< Edge< 3 > * > regina::Triangulation< 3 >::maximalForestInSkeleton | ( | bool | canJoinBoundaries = true | ) | const |
Produces a maximal forest in the triangulation's 1-skeleton.
It can be specified whether or not different boundary components may be joined by the maximal forest.
An edge leading to an ideal vertex is still a candidate for inclusion in the maximal forest. For the purposes of this algorithm, any ideal vertex will be treated as any other vertex (and will still be considered part of its own boundary component).
Note that the edge pointers returned will become invalid once the triangulation has changed.
- Parameters
-
canJoinBoundaries trueif and only if different boundary components are allowed to be joined by the maximal forest.
- Returns
- a set containing the edges of the maximal forest.
◆ meridianLongitude()
| std::pair< Edge< 3 > *, Edge< 3 > * > regina::Triangulation< 3 >::meridianLongitude | ( | ) |
Modifies a triangulated knot complement so that the meridian and algebraic longitude each follow a single boundary edge, and returns these two edges.
Assuming that this triangulation represents the complement of a knot in the 3-sphere, this routine:
- identifies the meridian of the knot complement, and also the algebraic longitude (i.e., the non-trivial simple closed curve on the boundary whose homology in the 3-manifold is trivial);
- layers additional tetrahedra on the boundary if necessary so that each of these curves is represented by a single boundary edge;
- returns these two (possibly new) boundary edges.
This routine uses fast heuristics to locate the meridian; as a result, it does not guarantee to terminate (but if you find a case where it does not, please let the Regina developers know!). If it does return then it guarantees that the result is correct.
Whilst this routine returns more information than longitude(), note that longitude() (1) runs much faster since it is based on fast algebraic calculations, and (2) guarantees to terminate.
At present this routine is fairly restrictive in what triangulations it can work with: it requires the triangulation to be one-vertex and have real (not ideal) boundary. These restrictions may be eased in future versions of Regina.
If the meridian and algebraic longitude are already both represented by single boundary edges, then it is guaranteed that this routine will not modify the triangulation, and will simply return these two boundary edges.
- Precondition
- The underlying 3-manifold is known to be the complement of a knot in the 3-sphere.
- This triangulation has precisely one vertex, and its (unique) boundary component is formed from two triangles.
- Warning
- This routine may modify the triangluation, as explained above, which will have the side-effect of invalidating any existing Vertex, Edge or Triangle references.
- If you have an ideal triangulation of a knot complement, you must first run idealToFinite() and then simplify the resulting triangulation to have two boundary triangles.
- Returns
- a pair (m, l), where m is the boundary edge representing the meridian and l is the boundary edge representing the algebraic longitude of the knot complement (after this triangulation has been modified if necessary). If an error (such as an integer overflow) occurs during the computation, then this routine will return (
null,null).
◆ minimiseBoundary()
| bool regina::Triangulation< 3 >::minimiseBoundary | ( | ) |
Ensures that the boundary contains the smallest possible number of triangles, potentially adding tetrahedra to do this.
This routine is for use with algorithms that require minimal boundaries (e.g., torus boundaries must contain exactly two triangles). As noted above, it may in fact increase the total number of tetrahedra in the triangulation (though the implementation does make efforts not to do this).
Once this routine is finished, every boundary component will have exactly one vertex, except for sphere and projective plane boundaries which will have exactly two triangles (but three and two vertices respectively).
The changes that this routine performs can always be expressed using only close book moves and/or layerings. In particular, this routine never creates new vertices, and it never creates a non-vertex-linking normal disc or sphere if there was not one before.
Although this routine only modifies real boundary components, it is fine if the triangulation also contains ideal boundary components (and these simply will be left alone).
If this triangulation is currently oriented, then this operation will preserve the orientation.
- Precondition
- This triangulation is valid.
- Returns
trueif the triangulation was changed, orfalseif every boundary component was already minimal to begin with.
◆ minimizeBoundary()
|
inline |
A synonym for minimiseBoundary().
This ensures that the boundary contains the smallest possible number of triangles, potentially adding tetrahedra to do this.
See minimiseBoundary() for further details.
- Precondition
- This triangulation is valid.
- Returns
trueif the triangulation was changed, orfalseif every boundary component was already minimal to begin with.
◆ moveContentsTo()
|
inherited |
Moves the contents of this triangulation into the given destination triangulation, without destroying any pre-existing contents.
All top-dimensional simplices that currently belong to dest will remain there (and will keep the same indices in dest). All top-dimensional simplices that belong to this triangulation will be moved into dest also (but in general their indices will change).
This triangulation will become empty as a result.
Any pointers or references to Simplex<dim> objects will remain valid.
If your intention is to replace the simplices in dest (i.e., you do not need to preserve the original contents), then consider using the move assignment operator instead (which is more streamlined and also moves across any cached properties from the source triangulation).
- Precondition
- dest is not this triangulation.
- Parameters
-
dest the triangulation into which simplices should be moved.
◆ newSimplex() [1/2]
|
inherited |
Creates a new top-dimensional simplex and adds it to this triangulation.
The new simplex will have an empty description. All (dim+1) facets of the new simplex will be boundary facets.
The new simplex will become the last simplex in this triangulation; that is, it will have index size()-1.
- Returns
- the new simplex.
◆ newSimplex() [2/2]
|
inherited |
Creates a new top-dimensional simplex with the given description and adds it to this triangulation.
All (dim+1) facets of the new simplex will be boundary facets.
Descriptions are optional, may have any format, and may be empty. How descriptions are used is entirely up to the user.
The new simplex will become the last simplex in this triangulation; that is, it will have index size()-1.
- Parameters
-
desc the description to give to the new simplex.
- Returns
- the new simplex.
◆ newSimplices() [1/2]
|
inherited |
Creates k new top-dimensional simplices, adds them to this triangulation, and returns them in a std::array.
The main purpose of this routine is to support structured binding; for example:
All new simplices will have empty descriptions, and all facets of each new simplex will be boundary facets.
The new simplices will become the last k simplices in this triangulation. Specifically, if the return value is the array ret, then each simplex ret[i] will have index size()-k+i in the overall triangulation.
- Python
- For Python users, the two variants of newSimplices() are essentially merged: the argument k is passed as an ordinary runtime argument, and the new top-dimensional simplices will be returned in a Python tuple of size k.
- Template Parameters
-
k the number of new top-dimensional simplices to add; this must be non-negative.
- Returns
- an array containing all of the new simplices, in the order in which they were added.
◆ newSimplices() [2/2]
|
inherited |
Creates k new top-dimensional simplices and adds them to this triangulation.
This is similar to the templated routine newSimplices<k>(), but with two key differences:
- This routine has the disadvantage that it does not return the new top-dimensional simplices, which means you cannot use it with structured binding.
- This routine has the advantage that k does not need to be known until runtime, which means this routine is accessible to Python users.
All new simplices will have empty descriptions, and all facets of each new simplex will be boundary facets.
The new simplices will become the last k simplices in this triangulation.
- Python
- For Python users, the two variants of newSimplices() are essentially merged: the argument k is passed as an ordinary runtime argument, and the new top-dimensional simplices will be returned in a Python tuple of size k.
- Parameters
-
k the number of new top-dimensional simplices to add; this must be non-negative.
◆ newTetrahedra() [1/2]
|
inline |
A dimension-specific alias for newSimplices().
See newSimplices() for further information.
◆ newTetrahedra() [2/2]
|
inline |
A dimension-specific alias for newSimplices().
See newSimplices() for further information.
◆ newTetrahedron() [1/2]
|
inline |
A dimension-specific alias for newSimplex().
See newSimplex() for further information.
◆ newTetrahedron() [2/2]
|
inline |
A dimension-specific alias for newSimplex().
See newSimplex() for further information.
◆ niceTreeDecomposition()
|
inline |
Returns a nice tree decomposition of the face pairing graph of this triangulation.
This can (for example) be used in implementing algorithms that are fixed-parameter tractable in the treewidth of the face pairing graph.
See TreeDecomposition for further details on tree decompositions, and see TreeDecomposition::makeNice() for details on what it means to be a nice tree decomposition.
This routine is fast: it will use a greedy algorithm to find a tree decomposition with (hopefully) small width, but with no guarantees that the width of this tree decomposition is the smallest possible.
The tree decomposition will be cached, so that if this routine is called a second time (and the underlying triangulation has not been changed) then the same tree decomposition will be returned immediately.
- Returns
- a nice tree decomposition of the face pairing graph of this triangulation.
◆ nonTrivialSphereOrDisc()
| std::optional< NormalSurface > regina::Triangulation< 3 >::nonTrivialSphereOrDisc | ( | ) | const |
Searches for a non-vertex-linking normal sphere or disc within this triangulation.
If such a surface exists within this triangulation, this routine is guaranteed to find one.
- Returns
- a non-vertex-linking normal sphere or disc, or no value if none exists.
◆ octagonalAlmostNormalSphere()
| std::optional< NormalSurface > regina::Triangulation< 3 >::octagonalAlmostNormalSphere | ( | ) | const |
Searches for an octagonal almost normal 2-sphere within this triangulation.
If such a surface exists, this routine is guaranteed to find one.
- Precondition
- This triangulation is valid, closed, orientable, connected, and 0-efficient. These preconditions are almost certainly more restrictive than they need to be, but we stay safe for now.
- Returns
- an octagonal almost normal 2-sphere, or no value if none exists.
◆ openBook()
| bool regina::Triangulation< 3 >::openBook | ( | Triangle< 3 > * | t, |
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a book opening move about the given triangle.
This involves taking a triangle meeting the boundary along two edges, and ungluing it to create two new boundary triangles (thus exposing the tetrahedra it initially joined). This move is the inverse of the closeBook() move, and is used to open the way for new shellBoundary() moves.
This move can be done if:
- the triangle meets the boundary in precisely two edges (and thus also joins two tetrahedra);
- the vertex between these two edges is a standard boundary vertex (its link is a disc);
- the remaining edge of the triangle (which is internal to the triangulation) is valid.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this operation will (trivially) preserve the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument f) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given triangle is a triangle of this triangulation.
- Parameters
-
t the triangle about which to perform the move. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ operator!=()
|
inlineinherited |
Determines if this triangulation is not combinatorially identical to the given triangulation.
Here "identical" means that the triangulations have the same number of top-dimensional simplices, with gluings between the same pairs of numbered simplices using the same gluing permutations. In other words, "identical" means that the triangulations are isomorphic via the identity isomorphism.
For the less strict notion of isomorphic triangulations, which allows relabelling of the top-dimensional simplices and their vertices, see isIsomorphicTo() instead.
This test does not examine the textual simplex descriptions, as seen in Simplex<dim>::description(); these may still differ. It also does not test whether lower-dimensional faces are numbered identically (vertices, edges and so on); this routine is only concerned with top-dimensional simplices.
(At the time of writing, two identical triangulations will always number their lower-dimensional faces in the same way. However, it is conceivable that in future versions of Regina there may be situations in which identical triangulations can acquire different numberings for vertices, edges, and so on.)
- Parameters
-
other the triangulation to compare with this.
- Returns
trueif and only if the two triangulations are not combinatorially identical.
◆ operator=() [1/2]
|
inline |
Sets this to be a (deep) copy of the given triangulation.
- Parameters
-
src the triangulation to copy.
- Returns
- a reference to this triangulation.
◆ operator=() [2/2]
|
inline |
Moves the contents of the given triangulation into this triangulation.
This is much faster than copy assignment, but is still linear time. This is because every tetrahedron must be adjusted to point back to this triangulation instead of src.
All tetrahedra and skeletal objects (faces, components and boundary components) that belong to src will be moved into this triangulation, and so any pointers or references to Tetrahedron<3>, Face<3, subdim>, Component<3> or BoundaryComponent<3> objects will remain valid. Likewise, all cached properties will be moved into this triangulation.
The triangulation that is passed (src) will no longer be usable.
- Note
- This operator is not marked
noexcept, since it fires change events on this triangulation which may in turn call arbitrary code via any registered packet listeners. It deliberately does not fire change events on src, since it assumes that src is about to be destroyed (which will fire a destruction event instead).
- Parameters
-
src the triangulation to move.
- Returns
- a reference to this triangulation.
◆ operator==()
|
inherited |
Determines if this triangulation is combinatorially identical to the given triangulation.
Here "identical" means that the triangulations have the same number of top-dimensional simplices, with gluings between the same pairs of numbered simplices using the same gluing permutations. In other words, "identical" means that the triangulations are isomorphic via the identity isomorphism.
For the less strict notion of isomorphic triangulations, which allows relabelling of the top-dimensional simplices and their vertices, see isIsomorphicTo() instead.
This test does not examine the textual simplex descriptions, as seen in Simplex<dim>::description(); these may still differ. It also does not test whether lower-dimensional faces are numbered identically (vertices, edges and so on); this routine is only concerned with top-dimensional simplices.
(At the time of writing, two identical triangulations will always number their lower-dimensional faces in the same way. However, it is conceivable that in future versions of Regina there may be situations in which identical triangulations can acquire different numberings for vertices, edges, and so on.)
In Regina 6.0.1 and earlier, this comparison was called isIdenticalTo().
- Parameters
-
other the triangulation to compare with this.
- Returns
trueif and only if the two triangulations are combinatorially identical.
◆ order()
| bool regina::Triangulation< 3 >::order | ( | bool | forceOriented = false | ) |
Relabels tetrahedron vertices in this triangulation to give an ordered triangulation, if possible.
To be an ordered triangulation, all face gluings (when restricted to the tetrahedron face) must be order preserving. In other words, it must be possible to orient all edges of the triangulation in such a fashion that they are consistent with the ordering of the vertices in each tetrahedron.
If it is possible to order this triangulation, the vertices of each tetrahedron will be relabelled accordingly and this routine will return true. Otherwise, this routine will return false and the triangulation will not be changed.
- Warning
- This routine may be slow, since it backtracks through all possible edge orientations until a consistent one has been found.
- Parameters
-
forceOriented trueif the triangulation must be both ordered and oriented, in which case this routine will returnfalseif the triangulation cannot be oriented and ordered at the same time. See orient() for further details.
- Returns
trueif the triangulation has been successfully ordered as described above, orfalseif not.
◆ orient()
|
inherited |
Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices are oriented consistently, if possible.
This routine works by flipping vertices (dim - 1) and dim of each top-dimensional simplices that has negative orientation. The result will be a triangulation where the top-dimensional simplices have their vertices labelled in a way that preserves orientation across adjacent facets. In particular, every gluing permutation will have negative sign.
If this triangulation includes both orientable and non-orientable components, the orientable components will be oriented as described above and the non-orientable components will be left untouched.
◆ pachner()
|
inherited |
Checks the eligibility of and/or performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face.
This involves replacing the (dim + 1 - k) top-dimensional simplices meeting that k-face with (k + 1) new top-dimensional simplices joined along a new internal (dim - k)-face. This can be done iff (i) the given k-face is valid and non-boundary; (ii) the (dim + 1 - k) top-dimensional simplices that contain it are distinct; and (iii) these simplices are joined in such a way that the link of the given k-face is the standard triangulation of the (dim - 1 - k)-sphere as the boundary of a (dim - k)-simplex.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal. In In the special case k = dim, the move is always legal and so the check argument will simply be ignored.
Note that after performing this move, all skeletal objects (facets, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument v) can no longer be used.
If this triangulation is currently oriented, then this Pachner move will label the new top-dimensional simplices in a way that preserves the orientation.
See the page on Pachner moves on triangulations for definitions and terminology relating to Pachner moves. After the move, the new belt face will be formed from vertices 0,1,...,(dim - k) of simplices().back().
- Warning
- For the case k = dim in Regina's standard dimensions, the labelling of the belt face has changed as of Regina 5.96 (the first prerelease for Regina 6.0). In versions 5.1 and earlier, the belt face was
simplices().back()->vertex(dim), and as of version 5.96 it is nowsimplices().back()->vertex(0).
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given k-face is a k-face of this triangulation.
- Parameters
-
f the k-face about which to perform the move. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
- Template Parameters
-
k the dimension of the given face. This must be between 0 and (dim) inclusive. You can still perform a Pachner move about a 0-face dim-face, but these moves use specialised implementations (as opposed to this generic template implementation).
◆ packet() [1/2]
|
inlineinherited |
Returns the packet that holds this data, if there is one.
If this object is being held by a packet p of type PacketOf<Held>, then that packet p will be returned. Otherwise, if this is a "standalone" object of type Held, then this routine will return null.
There is a special case when dealing with a packet q that holds a SnapPea triangulation. Here q is of type PacketOf<SnapPeaTriangulation>, and it holds a Triangulation<3> "indirectly" in the sense that Packetof<SnapPeaTriangulation> derives from SnapPeaTriangulation, which in turn derives from Triangulation<3>. In this scenario:
- calling Triangulation<3>::packet() will return
null, since there is no "direct" PacketOf<Triangulation<3>>; - calling SnapPeaTriangulation::packet() will return the enclosing packet q, since there is a PacketOf<SnapPeaTriangulation>;
- calling the special routine Triangulation<3>::inAnyPacket() will also return the "indirect" enclosing packet q.
The function inAnyPacket() is specific to Triangulation<3>, and is not offered for other Held types.
- Returns
- the packet that holds this data, or
nullif this data is not (directly) held by a packet.
◆ packet() [2/2]
|
inlineinherited |
Returns the packet that holds this data, if there is one.
See the non-const version of this function for further details, and in particular for how this functions operations in the special case of a packet that holds a SnapPea triangulation.
- Returns
- the packet that holds this data, or
nullif this data is not (directly) held by a packet.
◆ pentachora()
|
inlineinherited |
A dimension-specific alias for faces<4>(), or an alias for simplices() in dimension dim = 4.
This alias is available for dimensions dim ≥ 4.
See faces() for further information.
◆ pentachoron() [1/2]
|
inlineinherited |
◆ pentachoron() [2/2]
|
inlineinherited |
◆ pinchEdge()
| void regina::Triangulation< 3 >::pinchEdge | ( | Edge< 3 > * | e | ) |
Pinches an internal edge to a point.
Topologically, this collapses the edge to a point with no further side-effects, and it increases the number of tetrahedra by two.
This operation can be performed on any internal edge, without further constraints. Two particularly useful settings are:
- If the edge joins an internal vertex with some different vertex (which may be internal, boundary, ideal or invalid), then this move does not change the topology of the manifold at all, and it reduces the total number of vertices by one. In this sense, it acts as an alternative to collapseEdge(), and unlike collapseEdge() it can always be performed.
- If the edge runs from an internal vertex back to itself, then this move effectively drills out the edge, leaving an ideal torus or Klein bottle boundary component.
We do not allow e to lie entirely on the triangulation boundary, because the implementation actually collapses an internal curve parallel to e, not the edge e itself (and so if e is a boundary edge then the topological effect would not be as intended). We do allow e to be an internal edge with both endpoints on the boundary, but note that in this case the resulting topological operation would render the triangulation invalid.
If you are trying to reduce the number of vertices without changing the topology, and if e is an edge connecting an internal vertex with some different vertex, then either collapseEdge() or pinchEdge() may be more appropriate for your situation.
- The advantage of collapseEdge() is that it decreases the number of tetrahedra, whereas pinchEdge() increases this number (but only by two).
- The disadvantages of collapseEdge() are that it cannot always be performed, and its validity tests are expensive; pinchEdge() on the other hand can always be used for edges e of the type described above.
If this triangulation is currently oriented, then this operation will preserve the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.
- Precondition
- The given edge is an internal edge of this triangulation (that is, e does not lie entirely within the boundary).
- Parameters
-
e the edge to collapse.
◆ puncture()
| void regina::Triangulation< 3 >::puncture | ( | Tetrahedron< 3 > * | tet = nullptr | ) |
Punctures this manifold by removing a 3-ball from the interior of the given tetrahedron.
If no tetrahedron is specified (i.e., the tetrahedron pointer is null), then the puncture will be taken from the interior of tetrahedron 0.
The puncture will not meet the boundary of the tetrahedron, so nothing will go wrong if the tetrahedron has boundary facets and/or ideal vertices. A side-effect of this, however, is that the resulting triangulation will contain additional vertices, and will almost certainly be far from minimal. It is highly recommended that you run intelligentSimplify() if you do not need to preserve the combinatorial structure of the new triangulation.
The puncturing is done by subdividing the original tetrahedron. The new tetrahedra will have orientations consistent with the original tetrahedra, so if the triangulation was originally oriented then it will also be oriented after this routine has been called. See isOriented() for further details on oriented triangulations.
The new sphere boundary will be formed from two triangles; specifically, face 0 of the last and second-last tetrahedra of the triangulation. These two triangles will be joined so that vertex 1 of each tetrahedron coincides, and vertices 2,3 of one map to vertices 3,2 of the other.
- Precondition
- This triangulation is non-empty, and if
tetis non-null then it is in fact a tetrahedron of this triangulation.
- Parameters
-
tet the tetrahedron inside which the puncture will be taken. This may be null(the default), in which case the first tetrahedron will be used.
◆ recogniser() [1/2]
| std::string regina::Triangulation< 3 >::recogniser | ( | ) | const |
Returns a string that expresses this triangulation in Matveev's 3-manifold recogniser format.
Recogniser exports are currently not available for triangulations that are invalid or contain boundary triangles. If either of these conditions is true then this routine will throw an exception.
- Exceptions
-
NotImplemented This triangulation is either invalid or has boundary triangles.
- Returns
- a string containing the 3-manifold recogniser data.
◆ recogniser() [2/2]
| void regina::Triangulation< 3 >::recogniser | ( | std::ostream & | out | ) | const |
Writes a string expressing this triangulation in Matveev's 3-manifold recogniser format to the given output stream.
Recogniser exports are currently not available for triangulations that are invalid or contain boundary triangles. If either of these conditions is true then this routine will throw an exception.
- Exceptions
-
NotImplemented This triangulation is either invalid or has boundary triangles.
- Python
- Not present, but you can call recogniser() with no arguments which returns this data as a string.
- Parameters
-
out the output stream to which the recogniser data file will be written.
◆ recognizer() [1/2]
| std::string regina::Triangulation< 3 >::recognizer | ( | ) | const |
A synonym for recogniser().
This returns a string that expresses this triangulation in Matveev's 3-manifold recogniser format.
Recogniser exports are currently not available for triangulations that are invalid or contain boundary triangles. If either of these conditions is true then this routine will throw an exception.
- Exceptions
-
NotImplemented This triangulation is either invalid or has boundary triangles.
- Returns
- a string containing the 3-manifold recogniser data.
◆ recognizer() [2/2]
|
inline |
A synonym for recognizer(std::ostream&).
This writes a string expressing this triangulation in Matveev's 3-manifold recogniser format to the given output stream.
Recogniser exports are currently not available for triangulations that are invalid or contain boundary triangles. If either of these conditions is true then this routine will throw an exception.
- Exceptions
-
NotImplemented This triangulation is either invalid or has boundary triangles.
- Python
- Not present, but you can call recognizer() with no arguments which returns this data as a string.
- Parameters
-
out the output stream to which the recogniser data file will be written.
◆ reflect()
|
inherited |
Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices reflect their orientation.
In particular, if this triangulation is oriented, then it will be converted into an isomorphic triangulation with the opposite orientation.
This routine works by flipping vertices (dim - 1) and dim of every top-dimensional simplex.
◆ rehydrate()
|
static |
Rehydrates the given alphabetical string into a 3-dimensional triangulation.
For a full description of the dehydrated triangulation format, see A Census of Cusped Hyperbolic 3-Manifolds, Callahan, Hildebrand and Weeks, Mathematics of Computation 68/225, 1999.
The converse routine dehydrate() can be used to extract a dehydration string from an existing triangulation. Dehydration followed by rehydration might not produce a triangulation identical to the original, but it is guaranteed to produce an isomorphic copy. See dehydrate() for the reasons behind this.
- Exceptions
-
InvalidArgument the given string could not be rehydrated.
- Parameters
-
dehydration a dehydrated representation of the triangulation to construct. Case is irrelevant; all letters will be treated as if they were lower case.
- Returns
- the rehydrated triangulation.
◆ removeAllSimplices()
|
inlineinherited |
Removes all simplices from the triangulation.
As a result, this triangulation will become empty.
All of the simplices that belong to this triangulation will be destroyed immediately.
◆ removeAllTetrahedra()
|
inline |
A dimension-specific alias for removeAllSimplices().
See removeAllSimplices() for further information.
◆ removeSimplex()
|
inlineinherited |
Removes the given top-dimensional simplex from this triangulation.
The given simplex will be unglued from any adjacent simplices (if any), and will be destroyed immediately.
- Precondition
- The given simplex is a top-dimensional simplex in this triangulation.
- Parameters
-
simplex the simplex to remove.
◆ removeSimplexAt()
|
inlineinherited |
Removes the top-dimensional simplex at the given index in this triangulation.
This is equivalent to calling removeSimplex(simplex(index)).
The given simplex will be unglued from any adjacent simplices (if any), and will be destroyed immediately.
- Parameters
-
index specifies which top-dimensionalsimplex to remove; this must be between 0 and size()-1 inclusive.
◆ removeTetrahedron()
|
inline |
A dimension-specific alias for removeSimplex().
See removeSimplex() for further information.
◆ removeTetrahedronAt()
|
inline |
A dimension-specific alias for removeSimplexAt().
See removeSimplexAt() for further information.
◆ reorderTetrahedraBFS()
| void regina::Triangulation< 3 >::reorderTetrahedraBFS | ( | bool | reverse = false | ) |
Reorders the tetrahedra of this triangulation using a breadth-first search, so that small-numbered tetrahedra are adjacent to other small-numbered tetrahedra.
Specifically, the reordering will operate as follows. Tetrahedron 0 will remain tetrahedron 0. Its immediate neighbours will be numbered 1, 2, 3 and 4 (though if these neighbours are not distinct then of course fewer labels will be required). Their immediate neighbours will in turn be numbered 5, 6, and so on, ultimately following a breadth-first search throughout the entire triangulation.
If the optional argument reverse is true, then tetrahedron numbers will be assigned in reverse order. That is, tetrahedron 0 will become tetrahedron n-1, its neighbours will become tetrahedra n-2 down to n-5, and so on.
- Parameters
-
reverse trueif the new tetrahedron numbers should be assigned in reverse order, as described above.
◆ retriangulate()
|
inline |
Explores all triangulations that can be reached from this via Pachner moves, without exceeding a given number of additional tetrahedra.
Specifically, this routine will iterate through all triangulations that can be reached from this triangulation via 2-3 and 3-2 Pachner moves, without ever exceeding height additional tetrahedra beyond the original number.
For every such triangulation (including this starting triangulation), this routine will call action (which must be a function or some other callable object).
- action must take the following initial argument(s). Either (a) the first argument must be a triangulation (the precise type is discussed below), representing the triangluation that has been found; or else (b) the first two arguments must be of types const std::string& followed by a triangulation, representing both the triangulation and an isomorphism signature. The second form is offered in order to avoid unnecessary recomputation within the action function; however, note that the signature might not be of the IsoSigClassic type (i.e., it might not match the output from the default version of isoSig()). If there are any additional arguments supplied in the list args, then these will be passed as subsequent arguments to action.
- The triangulation argument will be passed as an rvalue; a typical action could (for example) take it by const reference and query it, or take it by value and modify it, or take it by rvalue reference and move it into more permanent storage.
- action must return a boolean. If action ever returns
true, then this indicates that processing should stop immediately (i.e., no more triangulations will be processed). - action may, if it chooses, make changes to this triangulation (i.e., the original triangulation upon which retriangulate() was called). This will not affect the search: all triangulations that this routine visits will be obtained via Pachner moves from the original form of this triangulation, before any subsequent changes (if any) were made.
- action will only be called once for each triangulation (including this starting triangulation). In other words, no triangulation will be revisited a second time in a single call to retriangulate().
This routine can be very slow and very memory-intensive, since the number of triangulations it visits may be superexponential in the number of tetrahedra, and it records every triangulation that it visits (so as to avoid revisiting the same triangulation again). It is highly recommended that you begin with height = 1, and if necessary try increasing height one at a time until this routine becomes too expensive to run.
If height is negative, then there will be no bound on the number of additional tetrahedra. This means that the routine will never terminate, unless action returns true for some triangulation that is passed to it.
Since Regina 7.0, this routine will not return until the exploration of triangulations is complete, regardless of whether a progress tracker was passed. If you need the old behaviour (where passing a progress tracker caused the enumeration to start in the background), simply call this routine in a new detached thread.
To assist with performance, this routine can run in parallel (multithreaded) mode; simply pass the number of parallel threads in the argument nThreads. Even in multithreaded mode, this routine will not return until processing has finished (i.e., either action returned true, or the search was exhausted). All calls to action will be protected by a mutex (i.e., different threads will never be calling action at the same time); as a corollary, the action should avoid expensive operations where possible (otherwise it will become a serialisation bottleneck in the multithreading).
- Precondition
- This triangulation is connected.
- Exceptions
-
FailedPrecondition this triangulation has more than one connected component. If a progress tracker was passed, it will be marked as finished before the exception is thrown.
- Warning
- The API for this class has not yet been finalised. This means that the class interface may change in new versions of Regina, without maintaining backward compatibility. If you use this class directly in your own code, please watch the detailed changelogs upon new releases to see if you need to make changes to your code.
- Python
- This function is available in Python, and the action argument may be a pure Python function. However, its form is more restricted: the arguments tracker and args are removed, so you call it as retriangulate(height, threads, action). Moreover, action must take exactly two arguments (const std::string&, Triangulation<3>&&) representing a signature and the triangulation, as described in option (b) above.
- Parameters
-
height the maximum number of additional tetrahedra to allow beyond the number of tetrahedra originally present in the triangulation, or a negative number if this should not be bounded. nThreads the number of threads to use. If this is 1 or smaller then the routine will run single-threaded. tracker a progress tracker through which progress will be reported, or nullif no progress reporting is required.action a function (or other callable object) to call for each triangulation that is found. args any additional arguments that should be passed to action, following the initial triangulation argument(s).
- Returns
trueif some call to action returnedtrue(thereby terminating the search early), orfalseif the search ran to completion.
◆ saveRecogniser()
| bool regina::Triangulation< 3 >::saveRecogniser | ( | const char * | filename | ) | const |
Writes this triangulation to the given file in Matveev's 3-manifold recogniser format.
Recogniser exports are currently not available for triangulations that are invalid or contain boundary triangles. If either of these conditions is true then the file will not be written, and this routine will return false.
- Internationalisation
- This routine makes no assumptions about the character encoding used in the given file name, and simply passes it through unchanged to low-level C/C++ file I/O routines. The contents of the file will be written using UTF-8.
- Parameters
-
filename the name of the Recogniser file to which to write.
- Returns
trueif and only if the file was successfully written.
◆ saveRecognizer()
|
inline |
A synonym for saveRecogniser().
This writes this triangulation to the given file in Matveev's 3-manifold recogniser format.
- Precondition
- This triangulation is not invalid, and does not contain any boundary triangles.
- Internationalisation
- This routine makes no assumptions about the character encoding used in the given file name, and simply passes it through unchanged to low-level C/C++ file I/O routines. The contents of the file will be written using UTF-8.
- Parameters
-
filename the name of the Recogniser file to which to write.
- Returns
trueif and only if the file was successfully written.
◆ saveSnapPea()
| bool regina::Triangulation< 3 >::saveSnapPea | ( | const char * | filename | ) | const |
Writes this triangulation to the given file using SnapPea's native file format.
Regarding what gets stored in the SnapPea data file:
- Since this function is defined by Regina's own Triangulation<3> class, only the tetrahedron gluings will be included in the SnapPea data file. All other SnapPea-specific information (such as peripheral curves) will be marked as unknown, since Regina does not track such information itself, and of course Regina-specific information (such as the Turaev-Viro invariants) will not be included in the SnapPea file either.
- The subclass SnapPeaTriangulation implements its own version of this function, which writes all additional SnapPea-specific information to the file (in fact it uses the SnapPea kernel itself to produce the file contents). However, to access that function you must explicitly call SnapPeaTriangulation::saveSnapPea() (since Triangulation<3> is not a polymorphic class, and in particular this function is not virtual).
SnapPea cannot represent triangulations that are empty, invalid, or contain boundary triangles. If any of these conditions is true then the file will not be written and this routine will return false.
- Internationalisation
- This routine makes no assumptions about the character encoding used in the given file name, and simply passes it through unchanged to low-level C/C++ file I/O routines. The contents of the file will be written using UTF-8.
- Parameters
-
filename the name of the SnapPea file to which to write.
- Returns
trueif and only if the file was successfully written.
◆ shellBoundary()
| bool regina::Triangulation< 3 >::shellBoundary | ( | Tetrahedron< 3 > * | t, |
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a boundary shelling move on the given tetrahedron.
This involves simply popping off a tetrahedron that touches the boundary. This can be done if:
- all edges of the tetrahedron are valid;
- precisely one, two or three faces of the tetrahedron lie in the boundary;
- if one face lies in the boundary, then the opposite vertex does not lie in the boundary, and no two of the remaining three edges are identified;
- if two faces lie in the boundary, then the remaining edge does not lie in the boundary, and the remaining two faces of the tetrahedron are not identified.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this operation will (trivially) preserve the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given tetrahedron is a tetrahedron of this triangulation.
- Parameters
-
t the tetrahedron upon which to perform the move. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ simplex() [1/2]
|
inlineinherited |
Returns the top-dimensional simplex at the given index in the triangulation.
Note that indexing may change when a simplex is added to or removed from the triangulation.
- Parameters
-
index specifies which simplex to return; this value should be between 0 and size()-1 inclusive.
- Returns
- the indexth top-dimensional simplex.
◆ simplex() [2/2]
|
inlineinherited |
Returns the top-dimensional simplex at the given index in the triangulation.
Note that indexing may change when a simplex is added to or removed from the triangulation.
- Parameters
-
index specifies which simplex to return; this value should be between 0 and size()-1 inclusive.
- Returns
- the indexth top-dimensional simplex.
◆ simplices()
|
inlineinherited |
Returns an object that allows iteration through and random access to all top-dimensional simplices in this triangulation.
The object that is returned is lightweight, and can be happily copied by value. The C++ type of the object is subject to change, so C++ users should use auto (just like this declaration does).
The returned object is guaranteed to be an instance of ListView, which means it offers basic container-like functions and supports C++11 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:
The object that is returned will remain up-to-date and valid for as long as the triangulation exists: even as simplices are added and/or removed, it will always reflect the simplices that are currently in the triangulation. Nevertheless, it is recommended to treat this object as temporary only, and to call simplices() again each time you need it.
- Returns
- access to the list of all top-dimensional simplices.
◆ simplifiedFundamentalGroup()
|
inlineinherited |
Notifies the triangulation that you have simplified the presentation of its fundamental group.
The old group presentation will be replaced by the (hopefully simpler) group that is passed.
This routine is useful for situations in which some external body (such as GAP) has simplified the group presentation better than Regina can.
Regina does not verify that the new group presentation is equivalent to the old, since this is - well, hard.
If the fundamental group has not yet been calculated for this triangulation, then this routine will store the new group as the fundamental group, under the assumption that you have worked out the group through some other clever means without ever having needed to call fundamentalGroup() at all.
Note that this routine will not fire a packet change event.
- Parameters
-
newGroup a new (and hopefully simpler) presentation of the fundamental group of this triangulation.
◆ simplifyExhaustive()
| bool regina::Triangulation< 3 >::simplifyExhaustive | ( | int | height = 1, |
| unsigned | nThreads = 1, |
||
| ProgressTrackerOpen * | tracker = nullptr |
||
| ) |
Attempts to simplify this triangulation using a slow but exhaustive search through the Pachner graph.
This routine is more powerful but much slower than intelligentSimplify().
Specifically, this routine will iterate through all triangulations that can be reached from this triangulation via 2-3 and 3-2 Pachner moves, without ever exceeding height additional tetrahedra beyond the original number.
If at any stage it finds a triangulation with fewer tetrahedra than the original, then this routine will call intelligentSimplify() to shrink the triangulation further if possible and will then return true. If it cannot find a triangulation with fewer tetrahedra then it will leave this triangulation unchanged and return false.
This routine can be very slow and very memory-intensive: the number of triangulations it visits may be superexponential in the number of tetrahedra, and it records every triangulation that it visits (so as to avoid revisiting the same triangulation again). It is highly recommended that you begin with height = 1, and if this fails then try increasing height one at a time until either you find a simplification or the routine becomes too expensive to run.
If height is negative, then there will be no bound on the number of additional tetrahedra. This means that the routine will not terminate until a simpler triangulation is found. If no simpler diagram exists then the only way to terminate this function is to cancel the operation via a progress tracker (read on for details).
If you want a fast simplification routine, you should call intelligentSimplify() instead. The benefit of simplifyExhaustive() is that, for very stubborn triangulations where intelligentSimplify() finds itself stuck at a local minimum, simplifyExhaustive() is able to "climb out" of such wells.
Since Regina 7.0, this routine will not return until either the triangulation is simplified or the exhaustive search is complete, regardless of whether a progress tracker was passed. If you need the old behaviour (where passing a progress tracker caused the exhaustive search to start in the background), simply call this routine in a new detached thread.
To assist with performance, this routine can run in parallel (multithreaded) mode; simply pass the number of parallel threads in the argument nThreads. Even in multithreaded mode, this routine will not return until processing has finished (i.e., either the triangulation was simplified or the search was exhausted).
If this routine is unable to simplify the triangulation, then the triangulation will not be changed.
- Precondition
- This triangulation is connected.
- Exceptions
-
FailedPrecondition this triangulation has more than one connected component. If a progress tracker was passed, it will be marked as finished before the exception is thrown.
- Python
- The global interpreter lock will be released while this function runs, so you can use it with Python-based multithreading.
- Parameters
-
height the maximum number of additional tetrahedra to allow beyond the number of tetrahedra originally present in the triangulation, or a negative number if this should not be bounded. nThreads the number of threads to use. If this is 1 or smaller then the routine will run single-threaded. tracker a progress tracker through which progress will be reported, or nullptrif no progress reporting is required.
- Returns
trueif and only if the triangulation was successfully simplified to fewer tetrahedra.
◆ simplifyToLocalMinimum()
| bool regina::Triangulation< 3 >::simplifyToLocalMinimum | ( | bool | perform = true | ) |
Uses all known simplification moves to reduce the triangulation monotonically to some local minimum number of tetrahedra.
End users will probably not want to call this routine. You should call intelligentSimplify() if you want a fast (and usually effective) means of simplifying a triangulation, or you should call simplifyExhaustive() if you are still stuck and you want to try a slower but more powerful method instead.
The moves used by this routine include 3-2, 2-0 (edge and vertex), 2-1 and boundary shelling moves.
Moves that do not reduce the number of tetrahedra (such as 4-4 moves or book opening moves) are not used in this routine. Such moves do however feature in intelligentSimplify().
If this triangulation is currently oriented, then this operation will preserve the orientation.
- Warning
- The implementation of this routine (and therefore its results) may change between different releases of Regina.
- Parameters
-
perform trueif we are to perform the simplifications, orfalseif we are only to investigate whether simplifications are possible (defaults totrue).
- Returns
- if perform is
true, this routine returnstrueif and only if the triangulation was changed to reduce the number of tetrahedra; if perform isfalse, this routine returnstrueif and only if it determines that it is capable of performing such a change.
◆ size()
|
inlineinherited |
Returns the number of top-dimensional simplices in the triangulation.
- Returns
- The number of top-dimensional simplices.
◆ snapPea() [1/2]
| std::string regina::Triangulation< 3 >::snapPea | ( | ) | const |
Returns a string containing the full contents of a SnapPea data file that describes this triangulation.
In particular, this string can be used in a Python session to pass the triangulation directly to SnapPy (without writing to the filesystem).
Regarding what gets stored in the SnapPea data file:
- Since this function is defined by Regina's own Triangulation<3> class, only the tetrahedron gluings will be included in the SnapPea data file. All other SnapPea-specific information (such as peripheral curves) will be marked as unknown, since Regina does not track such information itself, and of course Regina-specific information (such as the Turaev-Viro invariants) will not be included in the SnapPea file either.
- The subclass SnapPeaTriangulation implements its own version of this function, which writes all additional SnapPea-specific information to the file (in fact it uses the SnapPea kernel itself to produce the file contents). However, to access that function you must explicitly call SnapPeaTriangulation::snapPea() (since Triangulation<3> is not a polymorphic class, and in particular this function is not virtual).
If you wish to export a triangulation to a SnapPea file, you should call saveSnapPea() instead (which has better performance, and does not require you to construct an enormous intermediate string).
SnapPea cannot represent triangulations that are empty, invalid, or contain boundary triangles. If any of these conditions is true then this routine will throw an exception.
- Exceptions
-
NotImplemented This triangulation is either empty, invalid, or has boundary triangles.
- Returns
- a string containing the contents of the corresponding SnapPea data file.
◆ snapPea() [2/2]
| void regina::Triangulation< 3 >::snapPea | ( | std::ostream & | out | ) | const |
Writes the full contents of a SnapPea data file describing this triangulation to the given output stream.
Regarding what gets stored in the SnapPea data file:
- Since this function is defined by Regina's own Triangulation<3> class, only the tetrahedron gluings will be included in the SnapPea data file. All other SnapPea-specific information (such as peripheral curves) will be marked as unknown, since Regina does not track such information itself, and of course Regina-specific information (such as the Turaev-Viro invariants) will not be included in the SnapPea file either.
- The subclass SnapPeaTriangulation implements its own version of this function, which writes all additional SnapPea-specific information to the file (in fact it uses the SnapPea kernel itself to produce the file contents). However, to access that function you must explicitly call SnapPeaTriangulation::snapPea() (since Triangulation<3> is not a polymorphic class, and in particular this function is not virtual).
If you wish to extract the SnapPea data file as a string, you should call the zero-argument routine snapPea() instead. If you wish to write to a real SnapPea data file on the filesystem, you should call saveSnapPea() (which is also available in Python).
SnapPea cannot represent triangulations that are empty, invalid, or contain boundary triangles. If any of these conditions is true then this routine will throw an exception.
- Exceptions
-
NotImplemented This triangulation is either empty, invalid, or has boundary triangles.
- Python
- Not present, but you can call snapPea() with no arguments which returns this data as a string.
- Parameters
-
out the output stream to which the SnapPea data file will be written.
◆ str()
|
inherited |
Returns a short text representation of this object.
This text should be human-readable, should use plain ASCII characters where possible, and should not contain any newlines.
Within these limits, this short text ouptut should be as information-rich as possible, since in most cases this forms the basis for the Python str() and repr() functions.
- Python
- The Python "stringification" function
str()will use precisely this function, and for most classes the Pythonrepr()function will incorporate this into its output.
- Returns
- a short text representation of this object.
◆ strictAngleStructure()
|
inline |
Returns a strict angle structure on this triangulation, if one exists.
Recall that a strict angle structure is one in which every angle is strictly between 0 and π. If a strict angle structure does exist, then this routine is guaranteed to return one.
This routine is designed for scenarios where you already know that a strict angle structure exists. This means:
- If no strict angle structure exists, this routine will throw an exception, which will incur a significant overhead.
- If you do not know in advance whether a strict angle structure exists, you should call hasStrictAngleStructure() first. If the answer is no, this will avoid the overhead of throwing and catching exceptions. If the answer is yes, this will have the side-effect of caching the strict angle structure, which means your subsequent call to strictAngleStructure() will be essentially instantaneous.
The underlying algorithm runs a single linear program (it does not enumerate all vertex angle structures). This means that it is likely to be fast even for large triangulations.
The result of this routine is cached internally: as long as the triangulation does not change, multiple calls to strictAngleStructure() will return identical angle structures, and every call after the first be essentially instantaneous.
If the triangulation does change, however, then the cached angle structure will be deleted, and any reference that was returned before will become invalid.
- Exceptions
-
NoSolution no strict angle structure exists on this triangulation.
- Returns
- a strict angle structure on this triangulation, if one exists.
◆ summands()
| std::vector< Triangulation< 3 > > regina::Triangulation< 3 >::summands | ( | ) | const |
Computes the connected sum decomposition of this triangulation.
The prime summands will be returned as a vector of triangulations; this triangulation will not be modified.
As far as possible, the summands will be represented using 0-efficient triangulations (i.e., triangulations that contain no non-vertex-linking normal spheres). Specifically, for every summand, either:
- the triangulation of the summand that is produced will be 0-efficient; or
- the summand is one of RP3, the product S2xS1, or the twisted product S2x~S1. In each of these cases there is no possible 0-efficient triangulation of the summand, and so the triangulation that is produced will just be minimal (i.e., two tetrahedra).
For non-orientable triangulations, this routine is only guaranteed to succeed if the original manifold contains no embedded two-sided projective planes. If the manifold does contain embedded two-sided projective planes, then this routine might still succeed but it might fail; however, such a failure will always be detected, and in such a case this routine will throw an exception (as detailed below).
Note that this routine is currently only available for closed triangulations; see the list of preconditions for full details. If this triangulation is a 3-sphere then this routine will return an empty list.
This function is new to Regina 7.0, and it has some important changes of behaviour from the old connectedSumDecomposition() from Regina 6.0.1 and earlier:
- This function does not insert the resulting components into the packet tree.
- If this routine fails because of an embedded two-sided projective plane, then it throws an exception instead of returning -1.
- This function does not assign labels to the new summands.
The underlying algorithm appears in "A new approach to crushing 3-manifold triangulations", Discrete and Computational Geometry 52:1 (2014), pp. 116-139. This algorithm is based on the Jaco-Rubinstein 0-efficiency algorithm, and works in both orientable and non-orientable settings.
- Warning
- Users are strongly advised to check for exceptions if embedded two-sided projective planes are a possibility, since in such a case this routine might fail (as explained above). Note however that this routine might still succeed, and so success is not a proof that no embedded two-sided projective planes exist.
- The algorithms used in this routine rely on normal surface theory and so can be very slow for larger triangulations. For 3-sphere testing, see the routine isSphere() which uses faster methods where possible.
- Precondition
- This triangulation is valid, closed and connected.
- Exceptions
-
UnsolvedCase the original manifold is non-orientable and contains one or more embedded two-sided projective planes, and this routine was not able to recover from this situation.
- Returns
- a list of triangulations of prime summands.
◆ swap() [1/2]
|
inlineprotectednoexceptinherited |
Swap operation.
This should only be called when the entire type T contents of this object and other are being swapped. If one object has a current snapshot, then the other object will move in as the new image for that same snapshot. This avoids a deep copies of this object and/or other, even though both objects are changing.
In particular, if the swap function for T calls this base class function (as it should), then there is no need to call takeSnapshot() from either this object or src.
- Parameters
-
other the snapshot image being swapped with this.
◆ swap() [2/2]
| void regina::Triangulation< 3 >::swap | ( | Triangulation< 3 > & | other | ) |
Swaps the contents of this and the given triangulation.
All tetrahedra that belong to this triangulation will be moved to other, and all tetrahedra that belong to other will be moved to this triangulation. Likewise, all skeletal objects (such as lower-dimensional faces, components, and boundary components) and all cached properties (such as homology and fundamental group) will be swapped.
In particular, any pointers or references to Tetrahedron<3> and/or Face<3, subdim> objects will remain valid.
This routine will behave correctly if other is in fact this triangulation.
- Note
- This swap function is not marked
noexcept, since it fires change events on both triangulations which may in turn call arbitrary code via any registered packet listeners.
- Parameters
-
other the triangulation whose contents should be swapped with this.
◆ swapBaseData()
|
protectedinherited |
Swaps all data that is managed by this base class, including simplices, skeletal data, cached properties and the snapshotting data, with the given triangulation.
Note that TriangulationBase never calls this routine itself. Typically swapBaseData() is only ever called by Triangulation<dim>::swap().
- Parameters
-
other the triangulation whose data should be swapped with this.
◆ swapContents()
|
inline |
Deprecated routine that swaps the contents of this and the given triangulation.
- Deprecated:
- Use swap() instead.
- Parameters
-
other the triangulation whose contents should be swapped with this.
◆ takeSnapshot()
|
inlineprotectedinherited |
Must be called before modification and/or destruction of the type T contents.
See the Snapshot class notes for a full explanation of how this requirement works.
There are a few exceptions where takeSnapshot() does not need to be called: these are where type T move, copy and/or swap operations call the base class operations (as they should). See the Snapshottable move, copy and swap functions for details.
If this object has a current snapshot, then this function will trigger a deep copy with the snapshot.
After this function returns, this object is guaranteed to be completely unenrolled from the snapshotting machinery.
◆ tetrahedra()
|
inlineinherited |
A dimension-specific alias for faces<3>(), or an alias for simplices() in dimension dim = 3.
This alias is available for dimensions dim ≥ 3.
See faces() for further information.
◆ tetrahedron() [1/2]
|
inlineinherited |
◆ tetrahedron() [2/2]
|
inlineinherited |
◆ triangle() [1/2]
|
inlineinherited |
◆ triangle() [2/2]
|
inlineinherited |
◆ triangles()
|
inlineinherited |
A dimension-specific alias for faces<2>(), or an alias for simplices() in dimension dim = 2.
This alias is available for all dimensions.
See faces() for further information.
◆ triangulateComponents()
|
inherited |
Returns the individual connected components of this triangulation.
This triangulation will not be modified.
This function is new to Regina 7.0, and it has two important changes of behaviour from the old splitIntoComponents() from Regina 6.0.1 and earlier:
- This function does not insert the resulting components into the packet tree.
- This function does not assign labels to the new components.
- Returns
- a list of individual component triangulations.
◆ turaevViro()
| Cyclotomic regina::Triangulation< 3 >::turaevViro | ( | unsigned long | r, |
| bool | parity = true, |
||
| Algorithm | alg = ALG_DEFAULT, |
||
| ProgressTracker * | tracker = nullptr |
||
| ) | const |
Computes the given Turaev-Viro state sum invariant of this 3-manifold using exact arithmetic.
The initial data for the Turaev-Viro invariant is as described in the paper of Turaev and Viro, "State sum invariants of 3-manifolds and quantum 6j-symbols", Topology, vol. 31, no. 4, 1992, pp 865-902. In particular, Section 7 of this paper describes the initial data as determined by an integer r >= 3, and a root of unity q0 of degree 2r for which q0^2 is a primitive root of unity of degree r. There are several cases to consider:
- r may be even. In this case q0 must be a primitive (2r)th root of unity, and the invariant is computed as an element of the cyclotomic field of order 2r. There is no need to specify which root of unity is used, since switching between different roots of unity corresponds to an automorphism of the underlying cyclotomic field (i.e., it does not yield any new information). Therefore, if r is even, the additional argument parity is ignored.
- r may be odd, and q0 may be a primitive (2r)th root of unity. This case corresponds to passing the argument parity as
true. Here the invariant is again computed as an element of the cyclotomic field of order 2r. As before, there is no need to give further information as to which root of unity is used, since switching between roots of unity does not yield new information. - r may be odd, and q0 may be a primitive (r)th root of unity. This case corresponds to passing the argument parity as
false. In this case the invariant is computed as an element of the cyclotomic field of order r. Again, there is no need to give further information as to which root of unity is used.
This routine works entirely within the relevant cyclotomic field, which yields exact results but adds a significant overhead to the running time. If you want a fast floating-point approximation, you can call turaevViroApprox() instead.
Unlike this routine, turaevViroApprox() requires a precise specification of which root of unity is used (since it returns a numerical real value). The numerical value obtained by calling turaevViroApprox(r, whichRoot) should be the same as turaevViro(r, parity).evaluate(whichRoot), where parity is true or false according to whether whichRoot is odd or even respectively. Of course in practice the numerical values might be very different, since turaevViroApprox() performs significantly more floating point operations, and so is subject to a much larger potential numerical error.
If the requested Turaev-Viro invariant has already been computed, then the result will be cached and so this routine will be very fast (since it just returns the previously computed result). Otherwise the computation could be quite slow, particularly for larger triangulations and/or larger values of r.
Since Regina 7.0, this routine will not return until the Turaev-Viro computation is complete, regardless of whether a progress tracker was passed. If you need the old behaviour (where passing a progress tracker caused the computation to start in the background), simply call this routine in a new detached thread.
- Precondition
- This triangulation is valid, closed and non-empty.
- Python
- The global interpreter lock will be released while this function runs, so you can use it with Python-based multithreading.
- Parameters
-
r the integer r as described above; this must be at least 3. parity determines for odd r whether q0 is a primitive 2rth or rth root of unity, as described above. alg the algorithm with which to compute the invariant. If you are not sure, the default value (ALG_DEFAULT) is a safe choice. tracker a progress tracker through will progress will be reported, or nullptrif no progress reporting is required.
- Returns
- the requested Turaev-Viro invariant, or an uninitialised field element if the calculation was cancelled via the given progress tracker.
- See also
- allCalculatedTuraevViro
◆ turaevViroApprox()
| double regina::Triangulation< 3 >::turaevViroApprox | ( | unsigned long | r, |
| unsigned long | whichRoot = 1, |
||
| Algorithm | alg = ALG_DEFAULT |
||
| ) | const |
Computes the given Turaev-Viro state sum invariant of this 3-manifold using a fast but inexact floating-point approximation.
The initial data for the Turaev-Viro invariant is as described in the paper of Turaev and Viro, "State sum invariants of 3-manifolds and quantum 6j-symbols", Topology, vol. 31, no. 4, 1992, pp 865-902. In particular, Section 7 describes the initial data as determined by an integer r >= 3 and a root of unity q0 of degree 2r for which q0^2 is a primitive root of unity of degree r.
The argument whichRoot specifies which root of unity is used for q0. Specifically, q0 will be the root of unity e^(2i * Pi * whichRoot / 2r). There are additional preconditions on whichRoot to ensure that q0^2 is a primitive root of unity of degree r; see below for details.
This same invariant can be computed by calling turaevViro(r, parity).evaluate(whichRoot), where parity is true or false according to whether whichRoot is odd or even respectively. Calling turaevViroApprox() is significantly faster (since it avoids the overhead of working in cyclotomic fields), but may also lead to a much larger numerical error (since this routine might perform an exponential number of floating point operations, whereas the alternative only uses floating point for the final call to Cyclotomic::evaluate()).
These invariants, although computed in the complex field, should all be reals. Thus the return type is an ordinary double.
- Precondition
- This triangulation is valid, closed and non-empty.
- The argument whichRoot is strictly between 0 and 2r, and has no common factors with r.
- Parameters
-
r the integer r as described above; this must be at least 3. whichRoot specifies which root of unity is used for q0, as described above. alg the algorithm with which to compute the invariant. If you are not sure, the default value (ALG_DEFAULT) is a safe choice.
- Returns
- the requested Turaev-Viro invariant.
- See also
- allCalculatedTuraevViro
◆ twoOneMove()
| bool regina::Triangulation< 3 >::twoOneMove | ( | Edge< 3 > * | e, |
| int | edgeEnd, | ||
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a 2-1 move about the given edge.
This involves taking an edge meeting only one tetrahedron just once and merging that tetrahedron with one of the tetrahedra joining it.
This can be done assuming the following conditions:
- The edge must be valid and non-boundary.
- The two remaining faces of the tetrahedron are not joined, and the tetrahedron face opposite the given endpoint of the edge is not boundary.
- Consider the second tetrahedron to be merged (the one joined along the face opposite the given endpoint of the edge). Moreover, consider the two edges of this second tetrahedron that run from the (identical) vertices of the original tetrahedron not touching
eto the vertex of the second tetrahedron not touching the original tetrahedron. These edges must be distinct and may not both be in the boundary.
There are additional "distinct and not both boundary" conditions on faces of the second tetrahedron, but those follow automatically from the final condition above.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this 2-1 move will label the new tetrahedra in a way that preserves the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given edge is an edge of this triangulation.
- Parameters
-
e the edge about which to perform the move. edgeEnd the end of the edge opposite that at which the second tetrahedron (to be merged) is joined. The end is 0 or 1, corresponding to the labelling (0,1) of the vertices of the edge as described in EdgeEmbedding<3>::vertices(). check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ twoZeroMove() [1/2]
| bool regina::Triangulation< 3 >::twoZeroMove | ( | Edge< 3 > * | e, |
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a 2-0 move about the given edge of degree 2.
This involves taking the two tetrahedra joined at that edge and squashing them flat. This can be done if:
- the edge is valid and non-boundary;
- the two tetrahedra are distinct;
- the edges opposite
ein each tetrahedron are distinct and not both boundary; - if triangles f1 and f2 from one tetrahedron are to be flattened onto triangles g1 and g2 of the other respectively, then (a) f1 and g1 are distinct, (b) f2 and g2 are distinct, (c) we do not have both f1 = g2 and g1 = f2, (d) we do not have both f1 = f2 and g1 = g2, and (e) we do not have two of the triangles boundary and the other two identified.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this operation will preserve the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given edge is an edge of this triangulation.
- Parameters
-
e the edge about which to perform the move. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ twoZeroMove() [2/2]
| bool regina::Triangulation< 3 >::twoZeroMove | ( | Vertex< 3 > * | v, |
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a 2-0 move about the given vertex of degree 2.
This involves taking the two tetrahedra joined at that vertex and squashing them flat. This can be done if:
- the vertex is non-boundary and has a 2-sphere vertex link;
- the two tetrahedra are distinct;
- the triangles opposite
vin each tetrahedron are distinct and not both boundary; - the two tetrahedra meet each other on all three faces touching the vertex (as opposed to meeting each other on one face and being glued to themselves along the other two).
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this operation will preserve the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument v) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must be known in advance that the move is legal.
- The given vertex is a vertex of this triangulation.
- Parameters
-
v the vertex about which to perform the move. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check isfalse, the function simply returnstrue.
◆ utf8()
|
inherited |
Returns a short text representation of this object using unicode characters.
Like str(), this text should be human-readable, should not contain any newlines, and (within these constraints) should be as information-rich as is reasonable.
Unlike str(), this function may use unicode characters to make the output more pleasant to read. The string that is returned will be encoded in UTF-8.
- Returns
- a short text representation of this object.
◆ vertex()
|
inlineinherited |
◆ vertices()
|
inlineinherited |
A dimension-specific alias for faces<0>().
This alias is available for all dimensions dim.
See faces() for further information.
◆ writeTextLong()
|
inherited |
Writes a detailed text representation of this object to the given output stream.
- Python
- Not present; use detail() instead.
- Parameters
-
out the output stream to which to write.
◆ writeTextShort()
|
inherited |
Writes a short text representation of this object to the given output stream.
- Python
- Not present; use str() instead.
- Parameters
-
out the output stream to which to write.
◆ writeXMLBaseProperties()
|
protectedinherited |
Writes a chunk of XML containing properties of this triangulation.
This routine covers those properties that are managed by this base class TriangulationBase and that have already been computed for this triangulation.
This routine is typically called from within Triangulation<dim>::writeXMLPacketData(). The XML elements that it writes are child elements of the tri element.
- Parameters
-
out the output stream to which the XML should be written.
◆ zeroTwoMove() [1/3]
| bool regina::Triangulation< 3 >::zeroTwoMove | ( | Edge< 3 > * | e, |
| size_t | t0, | ||
| size_t | t1, | ||
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a 0-2 move about the (not necessarily distinct) triangles incident to e that are numbered t0 and t1.
This involves fattening up these two triangles into a new pair of tetrahedra around a new degree-two edge d; this is the inverse of performing a 2-0 move about the edge d. This can be done if and only if the following conditions are satisfied:
- The edge e is valid.
- The numbers t0 and t1 are both less than or equal to
e->degree(), and strictly less thane->degree()if e is non-boundary. This ensures that t0 and t1 correspond to sensible triangle numbers (as described below).
The triangles incident to e are numbered as follows:
- For each i from 0 up to
e->degree(), we assign the number i to the trianglee->embedding(i).tetrahedron()->triangle( e->embedding(i).vertices()[3] ) - If e is a boundary edge, then we additionally assign the number
e->degree()to the boundary trianglee->back().tetrahedron()->triangle( e->back().vertices()[2] )
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this 0-2 move will label the new tetrahedra in a way that preserves the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument e) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must by known in advance that the move is legal.
- The given edge e is an edge of this triangulation.
- Parameters
-
e the common edge of the two triangles about which to perform the move. t0 the number assigned to one of two triangles about which to perform the move. t1 the number assigned to the other triangle about which to perform the move. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returnstrue.
◆ zeroTwoMove() [2/3]
| bool regina::Triangulation< 3 >::zeroTwoMove | ( | EdgeEmbedding< 3 > | e0, |
| int | t0, | ||
| EdgeEmbedding< 3 > | e1, | ||
| int | t1, | ||
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a 0-2 move about the (not necessarily distinct) triangles e0.tetrahedron()->triangle( e0.vertices()[t0] ) and e1.tetrahedron()->triangle( e1.vertices()[t1] ).
This involves fattening up these two triangles into a new pair of tetrahedra around a new degree-two edge d; this is the inverse of performing a 2-0 move about the edge d. This can be done if and only if the following conditions are satisfied:
- e0 and e1 are both embeddings of the same edge e.
- t0 and t1 are both either 2 or 3; this ensures that the triangles about which we perform the move are triangles that are incident with e.
- The edge e is valid.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this 0-2 move will label the new tetrahedra in a way that preserves the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the arguments e0 and e1) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must by known in advance that the move is legal.
- The edge e is an edge of this triangulation.
- Parameters
-
e0 an embedding of the common edge e of the two triangles about which to perform the move. t0 one of the two triangles about which to perform the move (associated to the edge embedding e0). e1 another embedding of the edge e. t1 the other triangle about which to perform move (associated to the edge embedding e1). check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returnstrue.
◆ zeroTwoMove() [3/3]
| bool regina::Triangulation< 3 >::zeroTwoMove | ( | Triangle< 3 > * | t0, |
| int | e0, | ||
| Triangle< 3 > * | t1, | ||
| int | e1, | ||
| bool | check = true, |
||
| bool | perform = true |
||
| ) |
Checks the eligibility of and/or performs a 0-2 move about the (not necessarily distinct) triangles t0 and t1.
This involves fattening up these two triangles into a new pair of tetrahedra around a new degree-two edge d; this is the inverse of performing a 2-0 move about the edge d. This can be done if and only if the following conditions are satisfied:
- The edges
t0->edge(e0)andt1->edge(e1)are the same edge e of this triangulation. - The edge e is valid.
If the routine is asked to both check and perform, the move will only be performed if the check shows it is legal.
If this triangulation is currently oriented, then this 0-2 move will label the new tetrahedra in a way that preserves the orientation.
Note that after performing this move, all skeletal objects (triangles, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the arguments t0 and t1) can no longer be used.
- Precondition
- If the move is being performed and no check is being run, it must by known in advance that the move is legal.
- The given triangles t0 and t1 are triangles of this triangulation.
- The numbers e0 and e1 are both 0, 1 or 2.
- Parameters
-
t0 one of the two triangles about which to perform the move. e0 the edge at which t0 meets the other triangle about which to perform the move. t1 the other triangle about which to perform the move. e1 the edge at which t1 meets t0. check trueif we are to check whether the move is allowed (defaults totrue).perform trueif we are to perform the move (defaults totrue).
- Returns
- If check is
true, the function returnstrueif and only if the requested move may be performed without changing the topology of the manifold. If check is false, the function simply returnstrue.
Member Data Documentation
◆ boundaryComponents_
|
protectedinherited |
The components that form the boundary of the triangulation.
◆ compressingDisc_
| std::optional<bool> regina::Triangulation< 3 >::compressingDisc_ |
Does this 3-manifold contain a compressing disc?
◆ dimension
|
staticconstexprinherited |
A compile-time constant that gives the dimension of the triangulation.
◆ H1Bdry_
| std::optional<AbelianGroup> regina::Triangulation< 3 >::H1Bdry_ |
First homology group of the boundary.
◆ H1Rel_
| std::optional<AbelianGroup> regina::Triangulation< 3 >::H1Rel_ |
Relative first homology group of the triangulation with respect to the boundary.
◆ H2_
| std::optional<AbelianGroup> regina::Triangulation< 3 >::H2_ |
Second homology group of the triangulation.
◆ haken_
| std::optional<bool> regina::Triangulation< 3 >::haken_ |
Is this 3-manifold Haken? This property must only be stored for triangulations that are known to represent closed, connected, orientable, irreducible 3-manifolds.
◆ heldBy_
|
protectedinherited |
Indicates whether this Held object is in fact the inherited data for a PacketOf<Held>.
As a special case, this field is also used to indicate when a Triangulation<3> is in fact the inherited data for a SnapPeaTriangulation. See the PacketHeldBy enumeration for more details on the different values that this data member can take.
◆ irreducible_
| std::optional<bool> regina::Triangulation< 3 >::irreducible_ |
Is this 3-manifold irreducible?
◆ negativeIdealBoundaryComponents_
| std::optional<bool> regina::Triangulation< 3 >::negativeIdealBoundaryComponents_ |
Does the triangulation contain any boundary components that are ideal and have negative Euler characteristic?
◆ niceTreeDecomposition_
| std::optional<TreeDecomposition> regina::Triangulation< 3 >::niceTreeDecomposition_ |
A nice tree decomposition of the face pairing graph of this triangulation.
◆ simplices_
|
protectedinherited |
The top-dimensional simplices that form the triangulation.
◆ solidTorus_
| std::optional<bool> regina::Triangulation< 3 >::solidTorus_ |
Is this a triangulation of the solid torus?
◆ splittingSurface_
| std::optional<bool> regina::Triangulation< 3 >::splittingSurface_ |
Does the triangulation have a normal splitting surface?
◆ threeBall_
| std::optional<bool> regina::Triangulation< 3 >::threeBall_ |
Is this a triangulation of a 3-dimensional ball?
◆ threeSphere_
| std::optional<bool> regina::Triangulation< 3 >::threeSphere_ |
Is this a triangulation of a 3-sphere?
◆ topologyLock_
|
protectedinherited |
If non-zero, this will cause Triangulation<dim>::clearAllProperties() to preserve any computed properties that related to the manifold (as opposed to the specific triangulation).
This allows you to avoid recomputing expensive invariants when the underlying manifold is retriangulated.
This property should be managed by creating and destroying TopologyLock objects. The precise value of topologyLock_ indicates the number of TopologyLock objects that currently exist for this triangulation.
◆ turaevViroCache_
| TuraevViroSet regina::Triangulation< 3 >::turaevViroCache_ |
The set of Turaev-Viro invariants that have already been calculated.
See allCalculatedTuraevViro() for details.
◆ twoSphereBoundaryComponents_
| std::optional<bool> regina::Triangulation< 3 >::twoSphereBoundaryComponents_ |
Does the triangulation contain any 2-sphere boundary components?
◆ TxI_
| std::optional<bool> regina::Triangulation< 3 >::TxI_ |
Is this a triangulation of the product TxI?
◆ valid_
|
protectedinherited |
Is this triangulation valid? See isValid() for details on what this means.
◆ zeroEfficient_
| std::optional<bool> regina::Triangulation< 3 >::zeroEfficient_ |
Is the triangulation zero-efficient?
The documentation for this class was generated from the following file:
- triangulation/dim3/triangulation3.h