regina::Triangulation< dim > Class Template Reference

A dim-dimensional triangulation, built by gluing together dim-dimensional simplices along their (dim-1)-dimensional facets. More...

#include <triangulation/generic.h>

Inheritance diagram for regina::Triangulation< dim >:

Public Types
using	PacketChangeGroup
	A type alias for PacketChangeSpan, used when a span is being used purely for optimisation purposes.

Public Member Functions
bool	isReadOnlySnapshot () const
	Determines if this object is a read-only deep copy that was created by a snapshot.
std::shared_ptr< PacketOf< Triangulation< dim > > >	packet ()
	Returns the packet that holds this data, if there is one.
std::shared_ptr< const PacketOf< Triangulation< dim > > >	packet () const
	Returns the packet that holds this data, if there is one.
std::string	anonID () const
	A unique string ID that can be used in place of a packet ID.
std::string	str () const
	Returns a short text representation of this object.
std::string	utf8 () const
	Returns a short text representation of this object using unicode characters.
std::string	detail () const
	Returns a detailed text representation of this object.
std::string	tightEncoding () const
	Returns the tight encoding of this object.
size_t	hash () const
	Hashes this object to a non-negative integer, allowing it to be used for keys in hash tables.
Constructors and Destructors
	Triangulation ()
	Default constructor.
	Triangulation (const Triangulation &src)=default
	Creates a new copy of the given triangulation.
	Triangulation (const Triangulation &src, bool cloneProps, bool cloneLocks=true)
	Creates a new copy of the given triangulation, with the option of whether or not to clone its computed properties and/or locks also.
	Triangulation (Triangulation &&src) noexcept=default
	Moves the given triangulation into this new triangulation.
	~Triangulation ()
	Destroys this triangulation.
Simplices
size_t	size () const
	Returns the number of top-dimensional simplices in the triangulation.
auto	simplices () const
	Returns an object that allows iteration through and random access to all top-dimensional simplices in this triangulation.
Simplex< dim > *	simplex (size_t index)
	Returns the top-dimensional simplex at the given index in the triangulation.
const Simplex< dim > *	simplex (size_t index) const
	Returns the top-dimensional simplex at the given index in the triangulation.
Simplex< dim > *	newSimplex ()
	Creates a new top-dimensional simplex and adds it to this triangulation.
Simplex< dim > *	newSimplex (const std::string &desc)
	Creates a new top-dimensional simplex with the given description and adds it to this triangulation.
template<int k>
std::array< Simplex< dim > *, k >	newSimplices ()
	Creates k new top-dimensional simplices, adds them to this triangulation, and returns them in a std::array.
void	newSimplices (size_t k)
	Creates k new top-dimensional simplices and adds them to this triangulation.
void	removeSimplex (Simplex< dim > *simplex)
	Removes the given top-dimensional simplex from this triangulation.
void	removeSimplexAt (size_t index)
	Removes the top-dimensional simplex at the given index in this triangulation.
void	removeAllSimplices ()
	Removes all simplices from the triangulation.
void	moveContentsTo (Triangulation< dim > &dest)
	Moves the contents of this triangulation into the given destination triangulation, leaving this triangulation empty but otherwise usable.
bool	hasLocks () const
	Identifies whether any top-dimensional simplices and/or any of their facets are locked.
void	lockBoundary ()
	Locks all boundary facets of this triangulation.
void	unlockAll ()
	Unlocks all top-dimensional simplices and their facets.
Skeletal Queries
size_t	countComponents () const
	Returns the number of connected components in this triangulation.
size_t	countBoundaryComponents () const
	Returns the number of boundary components in this triangulation.
template<int subdim>
size_t	countFaces () const
	Returns the number of subdim-faces in this triangulation.
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.
size_t	countVertices () const
	A dimension-specific alias for countFaces<0>().
size_t	countEdges () const
	A dimension-specific alias for countFaces<1>().
size_t	countTriangles () const
	A dimension-specific alias for countFaces<2>().
size_t	countTetrahedra () const
	A dimension-specific alias for countFaces<3>().
size_t	countPentachora () const
	A dimension-specific alias for countFaces<4>().
std::vector< size_t >	fVector () const
	Returns the f-vector of this triangulation, which counts the number of faces of all dimensions.
auto	components () const
	Returns an object that allows iteration through and random access to all components of this triangulation.
auto	boundaryComponents () const
	Returns an object that allows iteration through and random access to all boundary components of this triangulation.
template<int subdim>
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.
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.
auto	vertices () const
	A dimension-specific alias for faces<0>().
auto	edges () const
	A dimension-specific alias for faces<1>().
auto	triangles () const
	A dimension-specific alias for faces<2>(), or an alias for simplices() in dimension dim = 2.
auto	tetrahedra () const
	A dimension-specific alias for faces<3>(), or an alias for simplices() in dimension dim = 3.
auto	pentachora () const
	A dimension-specific alias for faces<4>(), or an alias for simplices() in dimension dim = 4.
Component< dim > *	component (size_t index) const
	Returns the requested connected component of this triangulation.
BoundaryComponent< dim > *	boundaryComponent (size_t index) const
	Returns the requested boundary component of this triangulation.
template<int subdim>
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.
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.
Face< dim, 0 > *	vertex (size_t index) const
	A dimension-specific alias for face<0>().
Face< dim, 1 > *	edge (size_t index) const
	A dimension-specific alias for face<1>().
auto	triangle (size_t index)
	A dimension-specific alias for face<2>(), or an alias for simplex() in dimension dim = 2.
auto	triangle (size_t index) const
	A dimension-specific alias for face<2>(), or an alias for simplex() in dimension dim = 2.
auto	tetrahedron (size_t index)
	A dimension-specific alias for face<3>(), or an alias for simplex() in dimension dim = 3.
auto	tetrahedron (size_t index) const
	A dimension-specific alias for face<3>(), or an alias for simplex() in dimension dim = 3.
auto	pentachoron (size_t index)
	A dimension-specific alias for face<4>(), or an alias for simplex() in dimension dim = 4.
auto	pentachoron (size_t index) const
	A dimension-specific alias for face<4>(), or an alias for simplex() in dimension dim = 4.
template<int subdim>
Face< dim, subdim > *	translate (const Face< dim, subdim > *other) const
	Translates a face of some other triangulation into the corresponding face of this triangulation, using simplex numbers for the translation.
template<int subdim>
FaceEmbedding< dim, subdim >	translate (const FaceEmbedding< dim, subdim > &other) const
	Translates a face embedding from some other triangulation into the corresponding face embedding with respect to this triangulation, using simplex numbers for the translation.
FacetPairing< dim >	pairing () const
	Returns the dual graph of this triangulation, expressed as a facet pairing.
Basic Properties
bool	isEmpty () const
	Determines whether this triangulation is empty.
bool	isValid () const
	Determines if this triangulation is valid.
bool	hasBoundaryFacets () const
	Determines if this triangulation has any boundary facets.
size_t	countBoundaryFacets () const
	Returns the total number of boundary facets in this triangulation.
template<int subdim>
size_t	countBoundaryFaces () const
	Returns the number of boundary subdim-faces in this triangulation.
size_t	countBoundaryFaces (int subdim) const
	Returns the number of boundary subdim-faces in this triangulation, where the face dimension does not need to be known until runtime.
bool	isOrientable () const
	Determines if this triangulation is orientable.
bool	isConnected () const
	Determines if this triangulation is connected.
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.
long	eulerCharTri () const
	Returns the Euler characteristic of this triangulation.
Algebraic Properties
const GroupPresentation &	group () const
	Returns the fundamental group of this triangulation.
const GroupPresentation &	fundamentalGroup () const
	An alias for group(), which returns the fundamental group of this triangulation.
void	setGroupPresentation (GroupPresentation pres)
	Allows the specific presentation of the fundamental group to be changed by some other (external) means.
void	simplifiedFundamentalGroup (GroupPresentation pres)
	Deprecated alias for setGroupPresentation(), which allows the specific presentation of the fundamental group to be changed by some other (external) means.
template<int k = 1>
AbelianGroup	homology () const
	Returns the kth homology group of this triangulation, treating any ideal vertices as though they had been truncated.
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.
template<int k = 1>
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.
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.
template<int subdim>
MatrixInt	boundaryMap () const
	Returns the boundary map from subdim-faces to (subdim-1)-faces of the triangulation.
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.
template<int subdim>
MatrixInt	dualBoundaryMap () const
	Returns the boundary map from dual subdim-faces to dual (subdim-1)-faces of the triangulation.
MatrixInt	dualBoundaryMap (int subdim) const
	Returns the boundary map from dual subdim-faces to dual (subdim-1)-faces of the triangulation, where the face dimension does not need to be known until runtime.
template<int subdim>
MatrixInt	dualToPrimal () const
	Returns a map from dual chains to primal chains that preserves homology classes.
MatrixInt	dualToPrimal (int subdim) const
	Returns a map from dual chains to primal chains that preserves homology classes, where the chain dimension does not need to be known until runtime.
Skeletal Transformations
void	orient ()
	Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices are oriented consistently, if possible.
void	reflect ()
	Relabels the vertices of top-dimensional simplices in this triangulation so that all simplices reflect their orientation.
void	reorderBFS (bool reverse=false)
	Reorders the top-dimensional simplices of this triangulation using a breadth-first search, so that small-numbered simplices are adjacent to other small-numbered simplices.
Isomorphism< dim >	randomiseLabelling (bool preserveOrientation=true)
	Randomly relabels the top-dimensional simplices and their vertices.
template<int k>
bool	pachner (Face< dim, k > *f)
	If possible, performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face.
template<int k>
void	pachner (Face< dim, k > *f, Unprotected)
	Performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face, without any safety checks.
template<int k>
bool	pachner (Face< dim, k > *f, bool ignored, bool perform=true)
	Deprecated routine that tests for and optionally performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face of this triangulation.
template<int k>
bool	move20 (Face< dim, k > *f)
	If possible, performs a 2-0 move about the given k-face of degree two.
bool	shellBoundary (Simplex< dim > *s)
	If possible, performs a boundary shelling move upon the given top-dimensional simplex of this triangulation.
bool	shellBoundary (Simplex< dim > *s, bool ignored, bool perform=true)
	Deprecated routine that tests for and optionally performs a boundary shelling move on the given top-dimensional simplex.
template<int k>
bool	hasPachner (Face< dim, k > *f) const
	Determines whether it is possible to perform a (dim + 1 - k)-(k + 1) Pachner move about the given k-face of this triangulation, without violating any simplex and/or facet locks.
template<int k>
bool	has20 (Face< dim, k > *f) const
	Determines whether it is possible to perform a 2-0 move about the given k-face of this triangulation, without violating any simplex and/or facet locks.
bool	hasShellBoundary (Simplex< dim > *s) const
	Determines whether it is possible to perform a boundary shelling move upon the given top-dimensional simplex of this triangulation, without violating any simplex and/or facet locks.
template<int k>
std::optional< Triangulation< dim > >	withPachner (Face< dim, k > *f) const
	If possible, returns the triangulation obtained by performing a (dim + 1 - k)-(k + 1) Pachner move about the given k-face of this triangulation.
template<int k>
std::optional< Triangulation< dim > >	with20 (Face< dim, k > *f) const
	If possible, returns the triangulation obtained by performing a 2-0 move about the given k-face of this triangulation.
std::optional< Triangulation< dim > >	withShellBoundary (Simplex< dim > *s) const
	If possible, returns the triangulation obtained by performing a boundary shelling move on the given top-dimensional simplex of this triangulation.
template<int k>
bool	twoZeroMove (Face< dim, k > *f, bool ignored, bool perform=true)
	Deprecated routine that tests for and optionally performs a 2-0 move about the given k-face of this triangulation.
Subdivisions, Extensions and Covers
Triangulation< dim >	doubleCover () const
	Returns the orientable double cover of this triangulation.
Triangulation< dim >	doubleOverBoundary () const
	Returns two copies of this triangulation joined together along their boundary facets.
void	makeDoubleCover ()
	Deprecated routine that converts this triangulation into its orientable double cover.
void	subdivide ()
	Does a barycentric subdivision of the triangulation.
void	barycentricSubdivision ()
	Deprecated routine that performs a barycentric subdivision of the triangulation.
bool	makeIdeal ()
	Converts each real boundary component into a cusp (i.e., an ideal vertex).
bool	finiteToIdeal ()
	Alias for makeIdeal(), which converts each real boundary component into an ideal vertex.
Decompositions
std::vector< Triangulation< dim > >	triangulateComponents () const
	Returns the individual connected components of this triangulation.
Isomorphism Testing
bool	operator== (const TriangulationBase< dim > &other) const
	Determines if this triangulation is combinatorially identical to the given triangulation.
std::optional< Isomorphism< dim > >	isIsomorphicTo (const Triangulation< dim > &other) const
	Determines if this triangulation is combinatorially isomorphic to the given triangulation.
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).
template<typename Action , typename... Args>
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.
template<typename Action , typename... Args>
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).
bool	makeCanonical ()
	Relabel the top-dimensional simplices and their vertices so that this triangulation is in canonical form.
Building Triangulations
void	insertTriangulation (const Triangulation< dim > &source)
	Inserts a copy of the given triangulation into this triangulation.
void	insertTriangulation (Triangulation< dim > &&source)
	Moves the contents of the given triangulation into this triangulation.
Exporting Triangulations
void	writeTextShort (std::ostream &out) const
	Writes a short text representation of this object to the given output stream.
void	writeTextLong (std::ostream &out) const
	Writes a detailed text representation of this object to the given output stream.
template<class Type = IsoSigClassic<dim>, class Encoding = IsoSigPrintable<dim>>
Encoding::Signature	isoSig () const
	Constructs the isomorphism signature of the given type for this triangulation.
template<class Type = IsoSigClassic<dim>, class Encoding = IsoSigPrintable<dim>>
Encoding::Signature	sig () const
	Alias for isoSig(), which constructs the isomorphism signature of the given type for this triangulation.
template<class Type = IsoSigClassic<dim>, class Encoding = IsoSigPrintable<dim>>
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.
void	tightEncode (std::ostream &out) const
	Writes the tight encoding of this triangulation to the given output stream.
std::string	source (Language language=Language::Current) const
	Returns C++ or Python source code that can be used to reconstruct this triangulation.
std::string	dumpConstruction () const
	Deprecated routine that returns C++ code to reconstruct this triangulation.
void	writeDot (std::ostream &out, bool labels=false) const
	Writes the dual graph of this triangulation in the Graphviz DOT language.
std::string	dot (bool labels=false) const
	Returns a Graphviz DOT representation of the dual graph of this triangulation.

Static Public Member Functions
static Triangulation< dim >	tightDecoding (const std::string &enc)
	Reconstructs an object of type T from its given tight encoding.

Static Public Attributes
static constexpr int	dimension = dim
	A compile-time constant that gives the dimension of the triangulation.

Protected Member Functions
void	swap (Snapshottable &other) noexcept
	Swap operation.
void	takeSnapshot ()
	Must be called before modification and/or destruction of the type T contents.
bool	topologyLocked () const
	Returns whether or not there are any topology locks currently held on this object.

Protected Attributes
MarkedVector< Simplex< dim > >	simplices_
	The top-dimensional simplices that form the triangulation.
MarkedVector< Component< dim > >	components_
	The connected components that form the triangulation.
MarkedVector< BoundaryComponent< dim > >	boundaryComponents_
	The components that form the boundary of the triangulation.
std::array< size_t, dim >	nBoundaryFaces_
	The number of boundary faces of each dimension.
bool	valid_
	Is this triangulation valid? See isValid() for details on what this means.
PacketHeldBy	heldBy_
	Indicates whether this Held object is in fact the inherited data for a PacketOf<Held>.
uint8_t	topologyLock_ { 0 }
	The number of topology locks currently held on this object.

Simplices
class	detail::SimplexBase< dim >
class	detail::TriangulationBase< dim >
Triangulation &	operator= (const Triangulation &)=default
	Sets this to be a (deep) copy of the given triangulation.
Triangulation &	operator= (Triangulation &&src)=default
	Moves the contents of the given triangulation into this triangulation.
void	swap (Triangulation< dim > &other)
	Swaps the contents of this and the given triangulation.

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.
template<typename Iterator >
static Triangulation< dim >	fromGluings (size_t size, Iterator beginGluings, Iterator endGluings)
	Creates a triangulation from a list of gluings.
static Triangulation< dim >	fromIsoSig (const std::string &sig)
	Recovers a full triangulation from an isomorphism signature.
static Triangulation< dim >	fromSig (const std::string &sig)
	Alias for fromIsoSig(), to recover a full triangulation from an isomorphism signature.
static size_t	isoSigComponentSize (const std::string &sig)
	Deduces the number of top-dimensional simplices in a connected triangulation from its isomorphism signature.
static Triangulation< dim >	tightDecode (std::istream &input)
	Reconstructs a triangulation from its given tight encoding.
Simplex< dim > *	newSimplexRaw ()
	A variant of newSimplex() with no management of the underlying triangulation.
template<int k>
std::array< Simplex< dim > *, k >	newSimplicesRaw ()
	A variant of newSimplices() with no lock management, and no management of the underlying triangulation.
void	removeSimplexRaw (Simplex< dim > *simplex)
	A variant of removeSimplex() with no lock management, and no management of the underlying triangulation.
void	ensureSkeleton () const
	Ensures that all "on demand" skeletal objects have been calculated.
bool	calculatedSkeleton () const
	Determines whether the skeletal objects and properties of this triangulation have been calculated.
void	calculateSkeleton ()
	Calculates all skeletal objects for this triangulation.
void	cloneSkeleton (const TriangulationBase &src)
	Builds the skeleton of this triangulation as a clone of the skeleton of the given triangulation.
void	clearBaseProperties ()
	Clears all properties that are managed by this base class.
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.
void	writeXMLBaseProperties (std::ostream &out) const
	Writes a chunk of XML containing properties of this triangulation.

Detailed Description

template<int dim>
class regina::Triangulation< dim >

A dim-dimensional triangulation, built by gluing together dim-dimensional simplices along their (dim-1)-dimensional facets.

Typically (but not necessarily) such triangulations are used to represent dim-manifolds.

Structure of triangulations

Triangulations in Regina are not the same as pure simplicial complexes, for two reasons:

• The only identifications that the user can explicitly specify are gluings between dim-dimensional simplices along their (dim-1)-dimensional facets. All other identifications between k-faces (for any k) are simply consequences of these (dim-1)-dimensional gluings. In contrast, a simplicial complex allows explicit gluings between faces of any dimension.
• There is no requirement for a k-face to have (k+1) distinct vertices (so, for example, edges may be loops). Many distinct k-faces of a top-dimensional simplex may be identified together as a consequence of the (dim-1)-dimensional gluings, and indeed we are even allowed to glue together two distinct facets of the same dim-simplex. In contrast, a simplicial complex does not allow any of these situations.

Amongst other things, this definition is general enough to capture any reasonable definition of a dim-manifold triangulation. However, there is no requirement that a triangulation must actually represent a manifold (and indeed, testing this condition is undecidable for sufficiently large dim).

You can construct a triangulation from scratch using routines such as newSimplex() and Simplex<dim>::join(). There are also routines for exporting and importing triangulations in bulk, such as isoSig() and fromIsoSig() (which use isomorphism signatures), or source() and fromGluings() (which use C++ or Python code).

Skeleta and components

In additional to top-dimensional simplices, this class also tracks:

• connected components of the triangulation, as represented by the class Component<dim>;
• boundary components of the triangulation, as represented by the class BoundaryComponent<dim>;
• lower-dimensional faces of the triangulation, as represented by the classes Face<dim, subdim> for subdim = 0,...,(dim-1).

Such objects are temporary: whenever the triangulation changes, they will be deleted and rebuilt, and any pointers to them will become invalid. Likewise, if the triangulation is deleted then all component objects will be deleted alongside it.

The packet tree

Since Regina 7.0, this is no longer a "packet type" that can be inserted directly into the packet tree. Instead a Triangulation is now a standalone mathematatical object, which makes it slimmer and faster for ad-hoc use. The consequences of this are:

• If you create your own Triangulation, it will not have any of the usual packet infrastructure. You cannot add it into the packet tree, and it will not support a label, tags, child/parent packets, and/or event listeners.
• To include a Triangulation in the packet tree, you must create a new PacketOf<Triangulation>. This is a packet type, and supports labels, tags, child/parent packets, and event listeners. It derives from Triangulation, and so inherits the full Triangulation interface.

If you are adding new member functions that edit the internal data structures of a triangulation, you must remember to surround these changes with a ChangeAndClearSpan. This manages bookkeeping such as clearing computed properties, snapshotting, and (if this link does belong to a packet) firing packet change events.

C++ housekeeping

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.

For Regina's standard dimensions, this template is specialised and offers much more functionality. In order to use these specialised classes, you will need to include the corresponding headers (e.g., triangulation/dim2.h for dim = 2, or triangulation/dim3.h for dim = 3).

Python

Python does not support templates. Instead this class can be used by appending the dimension as a suffix (e.g., Triangulation2 and Triangulation3 for dimensions 2 and 3).

Template Parameters

dim	the dimension of the underlying triangulation. This must be between 2 and 15 inclusive.

Member Typedef Documentation

PacketChangeGroup

using regina::PacketData< Triangulation< dim > >::PacketChangeGroup

inherited

A type alias for PacketChangeSpan, used when a span is being used purely for optimisation purposes.

This type alias is used in the same way as Packet::PacketChangeGroup: it is purely for the benefit of the human reader, used to indicate that an event span is present purely for optimisation (and in particular, that the code would still be correct without it).

See Packet::PacketChangeGroup for further details.

Constructor & Destructor Documentation

Triangulation() [1/4]

template<int dim>

regina::Triangulation< dim >::Triangulation ( )

inline

Default constructor.

Creates an empty triangulation.

Triangulation() [2/4]

template<int dim>

regina::Triangulation< dim >::Triangulation ( const Triangulation< dim > & src )

default

Creates a new copy of the given triangulation.

This will also clone any computed properties (such as homology, fundamental group, and so on), as well as the skeleton (vertices, edges, components, etc.). In particular, the same numbering and labelling will be used for all skeletal objects.

If src has any locks on top-dimensional simplices and/or their facets, these locks will also be copied across.

If you want a "clean" copy that resets all properties to unknown, you can use the two-argument copy constructor instead.

Parameters

src	the triangulation to copy.

Triangulation() [3/4]

template<int dim>

regina::Triangulation< dim >::Triangulation ( const Triangulation< dim > & src, bool cloneProps, bool cloneLocks = true )

inline

Creates a new copy of the given triangulation, with the option of whether or not to clone its computed properties and/or locks also.

If cloneProps is true, then this constructor will also clone any computed properties (such as homology, fundamental group, and so on). If cloneProps is false, then these properties will be marked as unknown in the new triangulation, and will be recomputed on demand if/when they are required.

Regardless of cloneProps, the skeleton (vertices, edges, components, etc.) will always be cloned. This is to ensure that the same numbering and labelling will be used for all skeletal objects in both triangulations.

If cloneLocks is true then any locks on the top-dimensional simplices and/or facets of src will be copied across. If cloneLocks is false then the new triangulation will have no locks at all.

Parameters

src	the triangulation to copy.
cloneProps	true if this should also clone any computed properties of the given triangulation, or false if the new triangulation should have such properties marked as unknown.
cloneLocks	true if this should also clone any simplex and/or facet locks from the given triangulation, or false if the new triangulation should have no locks at all.

Triangulation() [4/4]

template<int dim>

regina::Triangulation< dim >::Triangulation ( Triangulation< dim > && src )

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 top-dimensional simplex must be adjusted to point back to this new triangulation instead of src.

All top-dimensional simplices 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 Simplex<dim>, Face<dim, subdim>, Component<dim> or BoundaryComponent<dim> objects will remain valid. Likewise, all cached properties will be moved into this triangulation.

If src has any locks on top-dimensional simplices and/or their facets, these locks will also be moved across.

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()

template<int dim>

regina::Triangulation< dim >::~Triangulation ( )

inline

Destroys this triangulation.

The constituent simplices, the cellular structure and all other properties will also be destroyed.

Member Function Documentation

anonID()

std::string regina::PacketData< Triangulation< dim > >::anonID ( ) const

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()

template<int dim>

void regina::detail::TriangulationBase< dim >::barycentricSubdivision ( )

inlineinherited

Deprecated routine that performs a barycentric subdivision of the triangulation.

Deprecated

This routine has been renamed to subdivide(), both to shorten the name but also to make it clearer that this triangulation will be modified directly.

Precondition

dim is one of Regina's standard dimensions.

Exceptions

LockViolation

This triangulation contains at least one locked top-dimensional simplex and/or facet. This exception will be thrown before any changes are made. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on how such locks work and what their implications are.

boundaryComponent()

template<int dim>

BoundaryComponent< dim > * regina::detail::TriangulationBase< dim >::boundaryComponent ( size_t index ) const

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()

template<int dim>

auto regina::detail::TriangulationBase< dim >::boundaryComponents ( ) const

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 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:

for (BoundaryComponent<dim>* b : tri.boundaryComponents()) { ... }

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]

template<int dim>

template<int subdim>

MatrixInt regina::detail::TriangulationBase< dim >::boundaryMap ( ) const

inherited

Returns the boundary map from subdim-faces to (subdim-1)-faces of the triangulation.

For C++ programmers who know subdim at compile time, you should use this template function boundaryMap<subdim>(), which is slightly faster than passing subdim as an ordinary runtime argument to boundaryMap(subdim).

See the non-templated boundaryMap(int) 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.

Python

Not present. Instead use the variant boundaryMap(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]

template<int dim>

MatrixInt regina::detail::TriangulationBase< dim >::boundaryMap ( int subdim ) const

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.

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.

For the boundary map, we fix orientations as follows. In simplicial homology, for any k, the orientation of a k-simplex is determined by assigning labels 0,...,k to its vertices. For this routine, since every k-face f is already a k-simplex, these labels will just be the inherent vertex labels 0,...,k of the corresponding Face<k> object. If you need to convert these labels into vertex numbers of a top-dimensional simplex containing f, you can use either Simplex<dim>::faceMapping<k>(), or the equivalent routine FaceEmbedding<k>::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.

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()

template<int dim>

bool regina::detail::TriangulationBase< dim >::calculatedSkeleton ( ) const

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

true if and only if the skeleton has been calculated.

calculateSkeleton()

template<int dim>

void regina::detail::TriangulationBase< dim >::calculateSkeleton ( )

protectedinherited

Calculates all skeletal objects for this triangulation.

For this parent class, calculateSkeleton() computes properties such as connected components, orientability, and lower-dimensional faces. Some Triangulation<dim> subclasses may track additional skeletal data, in which case they should reimplement this function. Their reimplementations must call this parent implementation.

You should never call this function directly; instead call ensureSkeleton() instead.

For developers: any data members that are computed and stored by calculateSkeleton() would typically also need to be cloned by cloneSkeleton(). Therefore any changes or extensions to calculateSkeleton() would typically need to come with analogous changes or extensions to cloneSkeleton() also.

Precondition

No skeletal objects have been computed, and the corresponding internal lists are all empty.

Warning

Any call to calculateSkeleton() must first cast down to Triangulation<dim>, to ensure that you are catching the subclass implementation if this exists. You should never directly call this parent implementation (unless of course you are reimplementing calculateSkeleton() in a Triangulation<dim> subclass).

clearBaseProperties()

template<int dim>

void regina::detail::TriangulationBase< dim >::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.

cloneSkeleton()

template<int dim>

void regina::detail::TriangulationBase< dim >::cloneSkeleton ( const TriangulationBase< dim > & src )

protectedinherited

Builds the skeleton of this triangulation as a clone of the skeleton of the given triangulation.

This clones all skeletal objects (e.g., faces, components and boundary components) and skeletal properties (e.g., validity and orientability). In general, this function clones the same properties and data that calculateSkeleton() computes.

For this parent class, cloneSkeleton() clones properties and data that are common to all dimensions. Some Triangulation<dim> subclasses may track additional skeletal properties or data, in which case they should reimplement this function (just as they also reimplement calculateSkeleton()). Their reimplementations must call this parent implementation.

This function is intended only for use by the copy constructor (and related "copy-like" constructors), and the copy assignment operator. Other code should typically not need to call this function directly.

The real point of this routine is to ensure that, when a triangulation is cloned, its skeleton is cloned with exactly the same numbering/labelling of its skeletal objects. To this end, it is fine to leave some "large" skeletal properties to be computed on demand where this is allowed (e.g., triangulated vertex links or triangulated boundary components, which are allowed to remain uncomputed until required, even when the full skeleton has been computed).

Precondition

No skeletal objects have been computed for this triangulation, and the corresponding internal lists are all empty.

The skeleton has been fully computed for the given source triangulation.

The given source triangulation is combinatorially identical to this triangulation (i.e., both triangulations have the same number of top-dimensional simplices, with gluings between the same pairs of numbered simplices using the same gluing permutations).

Warning

Any call to cloneSkeleton() must first cast down to Triangulation<dim>, to ensure that you are catching the subclass implementation if this exists. You should never directly call this parent implementation (unless of course you are reimplementing cloneSkeleton() in a Triangulation<dim> subclass).

Parameters

src	the triangulation whose skeleton should be cloned.

component()

template<int dim>

Component< dim > * regina::detail::TriangulationBase< dim >::component ( size_t index ) const

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()

template<int dim>

auto regina::detail::TriangulationBase< dim >::components ( ) const

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 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:

for (Component<dim>* c : tri.components()) { ... }

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.

countBoundaryComponents()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countBoundaryComponents ( ) const

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.

countBoundaryFaces() [1/2]

template<int dim>

template<int subdim>

size_t regina::detail::TriangulationBase< dim >::countBoundaryFaces ( ) const

inlineinherited

Returns the number of boundary subdim-faces in this triangulation.

This is the fastest way to count faces if you know subdim at compile time.

Specifically, this counts the number of subdim-faces for which isBoundary() returns true. This may lead to some unexpected results in non-standard scenarios; see the documentation for the non-templated countBoundaryFaces(int) for details.

Python

Not present. Instead use the variant countBoundaryFaces(subdim).

Template Parameters

subdim

the face dimension; this must be between 0 and dim-1 inclusive.

Returns

the number of boundary subdim-faces.

countBoundaryFaces() [2/2]

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countBoundaryFaces ( int subdim ) const

inlineinherited

Returns the number of boundary 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 countBoundaryFaces<subdim>() instead, which is fast constant time.

Specifically, this counts the number of subdim-faces for which isBoundary() returns true. This may lead to some unexpected results in non-standard scenarios; for example:

• In non-standard dimensions, ideal vertices are not recognised and so will not be counted as boundary;
• In an invalid triangulation, the number of boundary faces reported here may be smaller than the number of faces obtained when you triangulate the boundary using BoundaryComponent::build(). This is because "pinched" faces (where separate parts of the boundary are identified together) will only be counted once here, but will "spring apart" into multiple faces when the boundary is triangulated.

Exceptions

InvalidArgument

The face dimension subdim is outside the supported range (i.e., negative or greater than dim-1).

Parameters

subdim

the face dimension; this must be between 0 and dim-1 inclusive.

Returns

the number of boundary subdim-faces.

countBoundaryFacets()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countBoundaryFacets ( ) const

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.

This is equivalent to calling countBoundaryFaces<dim-1>().

Returns

the total number of boundary facets.

countComponents()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countComponents ( ) const

inlineinherited

Returns the number of connected components in this triangulation.

Returns

the number of connected components.

countEdges()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countEdges ( ) const

inlineinherited

A dimension-specific alias for countFaces<1>().

This alias is available for all dimensions dim.

See countFaces() for further information.

countFaces() [1/2]

template<int dim>

template<int subdim>

size_t regina::detail::TriangulationBase< dim >::countFaces ( ) const

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. 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]

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countFaces ( int subdim ) const

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()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countPentachora ( ) const

inlineinherited

A dimension-specific alias for countFaces<4>().

This alias is available for dimensions dim ≥ 4.

See countFaces() for further information.

countTetrahedra()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countTetrahedra ( ) const

inlineinherited

A dimension-specific alias for countFaces<3>().

This alias is available for dimensions dim ≥ 3.

See countFaces() for further information.

countTriangles()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countTriangles ( ) const

inlineinherited

A dimension-specific alias for countFaces<2>().

This alias is available for all dimensions dim.

See countFaces() for further information.

countVertices()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::countVertices ( ) const

inlineinherited

A dimension-specific alias for countFaces<0>().

This alias is available for all dimensions dim.

See countFaces() for further information.

detail()

std::string regina::Output< Triangulation< dim >, false >::detail ( ) const

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.

dot()

template<int dim>

std::string regina::detail::TriangulationBase< dim >::dot ( bool labels = false ) const

inlineinherited

Returns a Graphviz DOT representation of the dual graph of this triangulation.

Every vertex of this graph represents a top-dimensional simplex, and every edge represents a pair of simplex facets that are joined together. Note that for a closed triangulation this graph will be entirely (dim + 1)-valent; for triangulations with boundary facets, some graph vertices will have degree dim or less.

The output from this routine can be used as a standalone DOT file, ready for use with Graphviz. This DOT file will describe an undirected graph, and should be used with either the neato or fdp programs shipped with Graphviz.

The functions Triangulation<dim>::dot() and FacetPairing<dim>::dot() differ in a few ways:

• Triangulation<dim>::dot() does not support subgraph output (where you construct a large DOT file containing the dual graphs of many independent triangulations). If you need subgraph output, you can always call pairing().dot() instead.
• Triangulation<dim>::dot() makes more use of colour, in particular to indicate locked top-dimensional simplices and/or facets.

If you are simply writing this string to an output stream then you should call writeDot() instead, which is more efficient.

Parameters

labels

indicates whether graph vertices will be labelled with the corresponding simplex numbers.

Returns

the output of writeDot(), as outlined above.

doubleCover()

template<int dim>

Triangulation< dim > regina::detail::TriangulationBase< dim >::doubleCover ( ) const

inherited

Returns the orientable double cover of this triangulation.

Each orientable component will be duplicated, and each non-orientable component will be converted into its orientable double cover.

In general, the resulting double cover will not be oriented (though of course it will always be orientable).

If this triangulation has locks on any top-dimensional simplices and/or their facets, then these locks will be duplicated alongside their corresponding simplices and/or facets (i.e., they will appear in both sheets of the double cover).

Returns

the orientable double cover.

doubleOverBoundary()

template<int dim>

Triangulation< dim > regina::detail::TriangulationBase< dim >::doubleOverBoundary ( ) const

inherited

Returns two copies of this triangulation joined together along their boundary facets.

Specifically: the resulting triangulation is obtained by taking two copies S and T of this triangulation, and then gluing each boundary facet of S to the corresponding boundary facet of T using the identity permutation.

Any ideal vertices will be left alone (i.e., the ideal boundary components of S and T will not be joined together).

The resulting triangulation will not have any boundary facets. It will also not be oriented, even if this original triangulation is oriented, since S and T will be reflections of each other through the original triangulation boundary.

If this triangulation has no boundary facets (even if it does have ideal boundary components, then the result will simply be two disjoint copies of this original triangulation.

If this triangulation has locks on any top-dimensional simplices and/or their facets, then these locks will be duplicated alongside their corresponding simplices and/or facets (i.e., they will appear in both copies of the original triangulation). Any locks on boundary facets will not prevent this operation from completing successfully (i.e., they will not prevent the two copies S and T from being glued together).

Returns

two copies of this triangulation joined along their boundary facets.

dualBoundaryMap() [1/2]

template<int dim>

template<int subdim>

MatrixInt regina::detail::TriangulationBase< dim >::dualBoundaryMap ( ) const

inherited

Returns the boundary map from dual subdim-faces to dual (subdim-1)-faces of the triangulation.

For C++ programmers who know subdim at compile time, you should use this template function dualBoundaryMap<subdim>(), which is slightly faster than passing subdim as an ordinary runtime argument to dualBoundaryMap(subdim).

See the non-templated dualBoundaryMap(int) 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.

Python

Not present. Instead use the variant dualBoundaryMap(subdim).

Template Parameters

subdim

the dual face dimension; this must be between 1 and dim inclusive if dim is one of Regina's standard dimensions, or between 1 and (dim - 1) inclusive otherwise.

Returns

the boundary map from dual subdim-faces to dual (subdim-1)-faces.

dualBoundaryMap() [2/2]

template<int dim>

MatrixInt regina::detail::TriangulationBase< dim >::dualBoundaryMap ( int subdim ) const

inlineinherited

Returns the boundary map from dual subdim-faces to dual (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 dualBoundaryMap<subdim>() instead, which is slightly faster.

This function is analogous to boundaryMap(), but is designed to work with dual faces instead of ordinary (primal) faces. In particular, this is used in the implementation of homology(), which works with the dual skeleton in order to effectively truncate ideal vertices.

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 dual subdim-face of this triangulation, and the rth row corresponds to the rth dual (subdim-1)-face of this triangulation. Here we index dual faces in the same order as the (primal) faces of the triangulation that they are dual to, except that we omit primal boundary faces (i.e., primal faces for which Face::isBoundary() returns true). Therefore, for triangulations with boundary, the dual face indices and the corresponding primal face indices might not be equal.

For this dual boundary map, for positive dual face dimensions k, we fix the orientations of the dual k-faces as follows:

• In simplicial homology, the orientation of a k-simplex is determined by assigning labels 0,...,k to its vertices.
• Consider a dual k-face d, and let this be dual to the primal (dim-k)-face f. In general, d will not be a simplex. Let B denote the barycentre of f (which also appears as the "centre" point of d).
• Let emb be an arbitrary FaceEmbedding<dim-k> for f (i.e., chosen from f.embeddings()), and let s be the corresponding top-dimensional simplex containing f (i.e., emb.simplex()). For the special case of dual edges (k = 1), this choice matters; here we choose emb to be the first embedding (that is, f.front()). For larger k this choice does not matter; see below for the reasons why.
• Now consider how d intersects the top-dimensional simplex s. This intersection is a k-polytope with B as one of its vertices. We can extend this polytope away from B, pushing it all the way through the simplex s, until it becomes a k-simplex g whose vertices are B along with the k "unused" vertices of s that do not appear in f.
• We can now define the orientation of the dual k-face d to be the orientation of this k-simplex g that contains it. All that remains now is to orient g by choosing a labelling 0,...,k for its vertices.
• To orient g, we assign the label 0 to B, and we assign the labels 1,...,k to the "unused" vertices v[dim-k+1],...,v[dim] of s respectively, where v is the permutation emb.vertices().
• Finally, we note that for k > 1, the orientation for d does not depend on the particular choice of s and emb: by the preconditions and the fact that this routine only considers duals of non-boundary faces, the link of f must be a sphere, and therefore the images of those "other" vertices are fixed in a way that preserves orientation as you walk around the link. See the documentation for Simplex<dim>::faceMapping() for details.
• For the special case of dual edges (k = 1), the conditions above can be described more simply: the two endpoints of the dual edge d correspond to the two top-dimensional simplices on either side of the (dim-1)-face f, and we orient d by labelling these endpoints (0, 1) in the order (f.back(), f.front()).

If you wish to convert these boundary maps to homology groups yourself, either the AbelianGroup class (if you do not need to track which dual face is which) or the MarkedAbelianGroup class (if you do need to track individual dual faces) can help you do this.

Precondition

This triangulation is valid and non-empty.

Exceptions

InvalidArgument

The face dimension subdim is outside the supported range (as documented for the subdim argument below).

Parameters

subdim

the dual face dimension; this must be between 1 and dim inclusive if dim is one of Regina's standard dimensions, or between 1 and (dim - 1) inclusive otherwise.

Returns

the boundary map from dual subdim-faces to dual (subdim-1)-faces.

dualToPrimal() [1/2]

template<int dim>

template<int subdim>

MatrixInt regina::detail::TriangulationBase< dim >::dualToPrimal ( ) const

inherited

Returns a map from dual chains to primal chains that preserves homology classes.

For C++ programmers who know subdim at compile time, you should use this template function dualToPrimal<subdim>(), which is slightly faster than passing subdim as an ordinary runtime argument to dualToPrimal(subdim).

See the non-templated dualToPrimal(int) for full details on what this function computes and how the matrix it returns should be interpreted.

Precondition

This trianguation is valid, non-empty, and non-ideal. Note that Regina can only detect ideal triangulations in standard dimensions; for higher dimensions it is the user's reponsibility to confirm this some other way.

Python

Not present. Instead use the variant dualToPrimal(subdim).

Template Parameters

subdim

the chain dimension; this must be between 0 and (dim - 1) inclusive.

Returns

the map from dual subdim-chains to primal subdim-chains.

dualToPrimal() [2/2]

template<int dim>

MatrixInt regina::detail::TriangulationBase< dim >::dualToPrimal ( int subdim ) const

inlineinherited

Returns a map from dual chains to primal chains that preserves homology classes, where the chain 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 dualToPrimal<subdim>() instead, which is slightly faster.

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 dual subdim-face of this triangulation, and the rth row corresponds to the rth primal subdim-face of this triangulation.

We index and orient these dual and primal faces in the same manner as dualBoundaryMap() and boundaryMap() respectively. In particular, dual faces are indexed in the same order as the primal (dim-subdim)-faces of the triangulation that they are dual to, except that we omit primal boundary faces. See dualBoundaryMap() and boundaryMap() for further details.

The key feature of this map is that, if a column vector v represents a cycle c in the dual chain complex (i.e., it is a chain with zero boundary), and if this map is represented by the matrix M, then the vector M×v represents a cycle in the primal chain complex that belongs to the same subdimth homology class as c.

Regarding implementation: the map is constructed by (i) subdividing each dual face into smaller subdim-simplices whose vertices are barycentres of primal faces of different dimensions, (ii) moving each barycentre to vertex 0 of the corresponding face, and then (iii) discarding any resulting simplices with repeated vertices (which become "flattened" to a dimension less than subdim).

Precondition

This trianguation is valid, non-empty, and non-ideal. Note that Regina can only detect ideal triangulations in standard dimensions; for higher dimensions it is the user's reponsibility to confirm this some other way.

Exceptions

InvalidArgument

The chain dimension subdim is outside the supported range (as documented for the subdim argument below).

Parameters

subdim

the chain dimension; this must be between 0 and (dim - 1) inclusive.

Returns

the map from dual subdim-chains to primal subdim-chains.

dumpConstruction()

template<int dim>

std::string regina::detail::TriangulationBase< dim >::dumpConstruction ( ) const

inherited

Deprecated routine that returns C++ code to reconstruct this triangulation.

Deprecated

This is equivalent to calling source(Language::Cxx), for compatibility with older versions of Regina. In particular, it is not equivalent to calling source() (which defaults to the programming language currently being used). See source() for further details.

Returns

the C++ code that was generated.

edge()

template<int dim>

Face< dim, 1 > * regina::detail::TriangulationBase< dim >::edge ( size_t index ) const

inlineinherited

A dimension-specific alias for face<1>().

This alias is available for all dimensions dim.

See face() for further information.

edges()

template<int dim>

auto regina::detail::TriangulationBase< dim >::edges ( ) const

inlineinherited

A dimension-specific alias for faces<1>().

This alias is available for all dimensions dim.

See faces() for further information.

ensureSkeleton()

template<int dim>

void regina::detail::TriangulationBase< dim >::ensureSkeleton ( ) const

inlineprotectedinherited

Ensures that all "on demand" skeletal objects have been calculated.

eulerCharTri()

template<int dim>

long regina::detail::TriangulationBase< dim >::eulerCharTri ( ) const

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]

template<int dim>

auto regina::detail::TriangulationBase< dim >::face ( int subdim, size_t index ) const

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]

template<int dim>

template<int subdim>

Face< dim, subdim > * regina::detail::TriangulationBase< dim >::face ( size_t index ) const

inlineinherited

Returns the requested subdim-face of this triangulation, in a way that is optimised for C++ programmers.

Python

Not present. 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]

template<int dim>

template<int subdim>

auto regina::detail::TriangulationBase< dim >::faces ( ) const

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 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:

for (Face<dim, subdim>* f : tri.faces<subdim>()) { ... }

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. 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]

template<int dim>

auto regina::detail::TriangulationBase< dim >::faces ( int subdim ) const

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.

findAllIsomorphisms()

template<int dim>

template<typename Action , typename... Args>

bool regina::detail::TriangulationBase< dim >::findAllIsomorphisms ( const Triangulation< dim > & other, Action && action, Args &&... args ) const

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 of false indicates that the search for isomorphisms should continue, and a return value of true indicates 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 is findAllIsomorphisms(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

true if action ever terminated the search by returning true, or false if the search was allowed to run to completion.

findAllSubcomplexesIn()

template<int dim>

template<typename Action , typename... Args>

bool regina::detail::TriangulationBase< dim >::findAllSubcomplexesIn ( const Triangulation< dim > & other, Action && action, Args &&... args ) const

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 of false indicates that the search for isomorphisms should continue, and a return value of true indicates 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 is findAllSubcomplexesIn(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

true if action ever terminated the search by returning true, or false if the search was allowed to run to completion.

finiteToIdeal()

template<int dim>

bool regina::detail::TriangulationBase< dim >::finiteToIdeal ( )

inlineinherited

Alias for makeIdeal(), which converts each real boundary component into an ideal vertex.

This alias finiteToIdeal() is provided for compatibility with older versions of Regina. (It is not deprecated, and so this alias should remain part of Regina for a long time.)

See makeIdeal() for further details.

Exceptions

LockViolation

This triangulation contains at least one locked boundary facet. This exception will be thrown before any changes are made. See Simplex<dim>::lockFacet() for further details on how such locks work and what their implications are.

Returns

true if changes were made, or false if the original triangulation contained no real boundary components.

fromGluings() [1/2]

template<int dim>

template<typename Iterator >

Triangulation< dim > regina::detail::TriangulationBase< dim >::fromGluings ( size_t size, Iterator beginGluings, Iterator endGluings )

staticinherited

Creates a triangulation from a list of gluings.

This routine is an analogue to the variant of fromGluings() that takes a C++ 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:

tri = Triangulation3.fromGluings(2, [
    ( 0, 0, 1, Perm4(1,3,0,2) ), ( 0, 1, 1, Perm4(2,0,3,1) ),
    ( 0, 2, 1, Perm4(0,3,2,1) ), ( 0, 3, 1, Perm4(2,1,0,3) )])

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]

template<int dim>

Triangulation< dim > regina::detail::TriangulationBase< dim >::fromGluings ( size_t size, std::initializer_list< std::tuple< size_t, int, size_t, Perm< dim+1 > > > gluings )

inlinestaticinherited

Creates a triangulation from a hard-coded list of gluings.

This routine takes a C++ 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:

Triangulation<3> tri = Triangulation<3>::fromGluings(2, {
    { 0, 0, 1, {1,3,0,2} }, { 0, 1, 1, {2,0,3,1} },
    { 0, 2, 1, {0,3,2,1} }, { 0, 3, 1, {2,1,0,3} }});

Note

If you have an existing triangulation that you would like to hard-code in this way, you can call source() to generate the corresponding source code.

Exceptions

InvalidArgument

The given list of gluings does not correctly describe a triangulation with size top-dimensional simplices.

Python

Not present. Instead, use the variant of fromGluings() that takes this 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()

template<int dim>

static Triangulation< dim > regina::detail::TriangulationBase< dim >::fromIsoSig ( const std::string & sig )

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()

template<int dim>

Triangulation< dim > regina::detail::TriangulationBase< dim >::fromSig ( const std::string & sig )

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.

fundamentalGroup()

template<int dim>

const GroupPresentation & regina::detail::TriangulationBase< dim >::fundamentalGroup ( ) const

inlineinherited

An alias for group(), which returns the fundamental group of this triangulation.

See group() for further details, including how ideal vertices and invalid faces are managed.

Note

In Regina 7.2, the routine fundamentalGroup() was renamed to group() for brevity and for consistency with Link::group(). This more expressive name fundamentalGroup() will be kept as a long-term alias, and you are welcome to continue using it if you prefer.

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()

template<int dim>

std::vector< size_t > regina::detail::TriangulationBase< dim >::fVector ( ) const

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.

group()

template<int dim>

const GroupPresentation & regina::detail::TriangulationBase< dim >::group ( ) const

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 group() 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, group() should be called again; this will be instantaneous if the group has already been calculated.

Before Regina 7.2, this routine was called fundamentalGroup(). It has since been renamed to group() for brevity and for consistency with Link::group(). The more expressive name fundamentalGroup() will be kept, and you are welcome to use that instead if you prefer.

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.

has20()

template<int dim>

template<int k>

bool regina::detail::TriangulationBase< dim >::has20 ( Face< dim, k > * f ) const

inherited

Determines whether it is possible to perform a 2-0 move about the given k-face of this triangulation, without violating any simplex and/or facet locks.

For more detail on 2-0 moves and when they can be performed, see move20().

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be 0, 1 or 2, and must not exceed dim - 2.

Parameters

f	the k-face about which to perform the candidate move.

Returns

true if and only if the requested move can be performed.

hasBoundaryFacets()

template<int dim>

bool regina::detail::TriangulationBase< dim >::hasBoundaryFacets ( ) const

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

true if and only if there are boundary facets.

hash()

size_t regina::TightEncodable< Triangulation< dim > >::hash ( ) const

inlineinherited

Hashes this object to a non-negative integer, allowing it to be used for keys in hash tables.

This hash function makes use of Regina's tight encodings. In particular, any two objects with the same tight encoding will have equal hashes. This implementation (and therefore the specific hash value for each object) is subject to change in future versions of Regina.

Python

For Python users, this function uses the standard Python name hash(). This allows objects of this type to be used as keys in Python dictionaries and sets.

Returns

The integer hash of this object.

hasLocks()

template<int dim>

bool regina::detail::TriangulationBase< dim >::hasLocks ( ) const

inherited

Identifies whether any top-dimensional simplices and/or any of their facets are locked.

In short, locking a top-dimensional simplex and/or some of its facets means that that the simplex and/or facets must not be changed. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for full details on how locks work and what their implications are.

Returns

true if and only if there is at least one locked top-dimensional simplex or at least one locked facet of a top-dimensional simplex within this triangulation.

hasPachner()

template<int dim>

template<int k>

bool regina::detail::TriangulationBase< dim >::hasPachner ( Face< dim, k > * f ) const

inherited

Determines whether it is possible to perform a (dim + 1 - k)-(k + 1) Pachner move about the given k-face of this triangulation, without violating any simplex and/or facet locks.

For more detail on Pachner moves and when they can be performed, see pachner().

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be between 0 and (dim) inclusive.

Parameters

f	the k-face about which to perform the candidate move.

Returns

true if and only if the requested move can be performed.

hasShellBoundary()

template<int dim>

bool regina::detail::TriangulationBase< dim >::hasShellBoundary ( Simplex< dim > * s ) const

inherited

Determines whether it is possible to perform a boundary shelling move upon the given top-dimensional simplex of this triangulation, without violating any simplex and/or facet locks.

This test is only available in standard dimensions, since Regina's notion of "valid faces" is weaker in higher dimensions (due to the need to solve undecidable problems). See Face::isValid() for further discussion.

For more detail on boundary shelling moves and when they can be performed, see shellBoundary().

Precondition

The dimension dim is one of Regina's standard dimensions.

The given simplex is a simplex of this triangulation.

Parameters

s	the top-dimensional simplex upon which to perform the candidate move.

Returns

true if and only if the requested move can be performed.

homology() [1/2]

template<int dim>

template<int k = 1>

AbelianGroup regina::detail::TriangulationBase< dim >::homology ( ) const

inherited

Returns the kth homology group of this triangulation, treating any ideal vertices as though they had been truncated.

For C++ programmers who know subdim at compile time, you should use this template function homology<subdim>(), which is slightly faster than passing subdim as an ordinary runtime argument to homology(subdim).

See the non-templated homology(int) 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.

Python

Not present. 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]

template<int dim>

AbelianGroup regina::detail::TriangulationBase< dim >::homology ( int k ) const

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.

A problem with computing homology 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.
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.

insertTriangulation() [1/2]

template<int dim>

void regina::detail::TriangulationBase< dim >::insertTriangulation ( const Triangulation< dim > & source )

inherited

Inserts a copy of the given triangulation into this triangulation.

The top-dimensional simplices of source will be copied into this triangulation, and placed after any pre-existing simplices. Specifically, if the original size of this triangulation was N, then source.simplex(i) will be copied to a new simplex which will appear as simplex(N+i) of this triangulation.

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.

As a trivial consequence, if this and the given triangulation are both oriented, then the result will preserve these orientations.

If source has locks on any top-dimensional simplices and/or their facets, these locks will also be copied over to this triangulation.

This routine behaves correctly when source is this triangulation.

Parameters

source

the triangulation whose copy will be inserted.

insertTriangulation() [2/2]

template<int dim>

void regina::detail::TriangulationBase< dim >::insertTriangulation ( Triangulation< dim > && source )

inherited

Moves the contents of the given triangulation into this triangulation.

The top-dimensional simplices of source will be moved directly into this triangulation, and placed after any pre-existing simplices. Specifically, if the original size of this triangulation was N, then source.simplex(i) will become simplex(N+i) of this triangulation.

As is normal for an rvalue reference, after calling this function source will be unusable. Any simplex pointers that referred to either this triangulation or source will remain valid (and will all now refer to this triangulation), though if they originally referred to source then they will now return different indices. Any locks on top-dimensional simplices and/or their facets will be preserved.

Calling tri.insertTriangulation(source) (where source is an rvalue reference) is similar to calling source.moveContentsTo(tri), but it is a little faster since it does not need to leave source in a usable state.

Regarding packet change events: this function does not fire a change event on source, since it assumes that source is about to be destroyed (which will fire a destruction event instead).

Precondition

source is not this triangulation.

Python

Not present. Only the copying version of this function is available (i.e., the version that takes source as a const reference). If you want a fast move operation, call source.moveContentsTo(this).

Parameters

source

the triangulation whose contents should be moved.

isConnected()

template<int dim>

bool regina::detail::TriangulationBase< dim >::isConnected ( ) const

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

true if and only if this triangulation is connected.

isContainedIn()

template<int dim>

std::optional< Isomorphism< dim > > regina::detail::TriangulationBase< dim >::isContainedIn ( const Triangulation< dim > & other ) const

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.

Like isIsomorphicTo() and the equality test, this routine does not examine or compare simplex/facet locks.

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()

template<int dim>

bool regina::detail::TriangulationBase< dim >::isEmpty ( ) const

inlineinherited

Determines whether this triangulation is empty.

An empty triangulation is one with no simplices at all.

Returns

true if and only if this triangulation is empty.

isIsomorphicTo()

template<int dim>

std::optional< Isomorphism< dim > > regina::detail::TriangulationBase< dim >::isIsomorphicTo ( const Triangulation< dim > & other ) const

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 the equality test t1 == t2.

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, like the equality test, this routine does not examine or compare simplex/facet locks.

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 use the stricter equality test t1 == t2 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.

isOrientable()

template<int dim>

bool regina::detail::TriangulationBase< dim >::isOrientable ( ) const

inlineinherited

Determines if this triangulation is orientable.

Returns

true if and only if this triangulation is orientable.

isOriented()

template<int dim>

bool regina::detail::TriangulationBase< dim >::isOriented ( ) const

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

true if and only if all top-dimensional simplices are oriented consistently.

Author

Matthias Goerner

isoSig()

template<int dim>

template<class Type = IsoSigClassic<dim>, class Encoding = IsoSigPrintable<dim>>

Encoding::Signature regina::detail::TriangulationBase< dim >::isoSig ( ) const

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.

By default, isomorphism signatures support simplex and facet locks: this means that locks are encoded in the isomorphism signature, and the "combinatorial isomorphisms" mentioned above must respect not just the gluings between simplices but also any simplex and/or facet locks. You can change this behaviour by requesting a different encoding (such as IsoSigPrintableLockFree). Under the default encoding, if this triangulation does not have any simplex and/or facet locks then the isomorphism signature will look the same as it did in Regina 7.3.x and earlier, before locks were supported.

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!) n² log² 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 (including any simplex and/or facet locks) 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.
• If you wish to produce an isomorphism signature that ignores simplex and/or facet locks then you can use an encoding whose encode() function ignores the final locks argument, such as IsoSigPrintableLockFree.

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. To use the default signature type and encoding, just call isoSig(). To use a non-default signature type, add a suffix _Type where Type is an abbreviated version of the signature type (e.g., isoSig_EdgeDegrees() for the signature type IsoSigEdgeDegrees). To use the encoding IsoSigPrintableLockFree (the only non-default encoding available at present), add another suffix _LockFree (e.g., isoSig_LockFree(), or isoSig_EdgeDegrees_LockFree()).

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()

template<int dim>

static size_t regina::detail::TriangulationBase< dim >::isoSigComponentSize ( const std::string & sig )

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()

template<int dim>

template<class Type = IsoSigClassic<dim>, class Encoding = IsoSigPrintable<dim>>

std::pair< typename Encoding::Signature, Isomorphism< dim > > regina::detail::TriangulationBase< dim >::isoSigDetail ( ) const

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(this).

Python

Although this is a templated function, all of the variants supplied with Regina are available to Python users. To use the default signature type and encoding, just call isoSigDetail(). To use a non-default signature type, add a suffix _Type where Type is an abbreviated version of the signature type (e.g., isoSigDetail_EdgeDegrees() for the signature type IsoSigEdgeDegrees). To use the encoding IsoSigPrintableLockFree (the only non-default encoding available at present), add another suffix _LockFree (e.g., isoSigDetail_LockFree(), or isoSigDetail_EdgeDegrees_LockFree()).

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()

bool regina::Snapshottable< Triangulation< dim > >::isReadOnlySnapshot ( ) const

inlineinherited

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.)

Returns

true if and only if this object is a deep copy that was taken by a Snapshot object of some original type T image.

isValid()

template<int dim>

bool regina::detail::TriangulationBase< dim >::isValid ( ) const

inlineinherited

Determines if this triangulation is valid.

There are several conditions that might make a dim-dimensional triangulation invalid:

1. 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);
2. 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

true if and only if this triangulation is valid.

lockBoundary()

template<int dim>

void regina::detail::TriangulationBase< dim >::lockBoundary ( )

inherited

Locks all boundary facets of this triangulation.

In short, this means that the boundary facets must not be changed. See Simplex<dim>::lockFacet() for full details on how locks work and what their implications are.

If there are any other locks on top-dimensional simplices and/or their facets, these other locks will be left intact.

Note that this only locks the facets of real boundary components. Ideal boundary components are not affected (since they have no facets to lock).

makeCanonical()

template<int dim>

bool regina::detail::TriangulationBase< dim >::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).

If this triangulation has any locks on its top-dimensional simplices and/or their facets, this routine will carry the locks through the relabelling correctly. Locks do not play any role in determining which labelling is canonical (i.e., the canonical labelling will be the same regardles of whether or not there are locks present).

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

true if the triangulation was changed, or false if the triangulation was in canonical form to begin with.

makeDoubleCover()

template<int dim>

void regina::detail::TriangulationBase< dim >::makeDoubleCover ( )

inherited

Deprecated routine that converts this triangulation into its orientable double cover.

This triangulation wll be modified directly.

Deprecated

This routine has been replaced by doubleCover(), which returns the result as a new triangulation and leaves the original triangulation untouched.

See doubleCover() for further details.

makeIdeal()

template<int dim>

bool regina::detail::TriangulationBase< dim >::makeIdeal ( )

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 a truncateIdeal() 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).

A note: this operation does not preserve orientedness. That is, even if this triangulation was oriented before calling this function, it might not be oriented after. This behaviour may change in a future version of Regina.

Warning

If a real boundary component contains vertices whose links are not discs, this operation may have unexpected results.

Exceptions

LockViolation

This triangulation contains at least one locked boundary facet. This exception will be thrown before any changes are made. See Simplex<dim>::lockFacet() for further details on how such locks work and what their implications are.

Returns

true if changes were made, or false if the original triangulation contained no real boundary components.

markedHomology() [1/2]

template<int dim>

template<int k>

MarkedAbelianGroup regina::detail::TriangulationBase< dim >::markedHomology ( ) const

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.

For C++ programmers who know subdim at compile time, you should use this template function markedHomology<subdim>(), which is slightly faster than passing subdim as an ordinary runtime argument to markedHomology(subdim).

See the non-templated markedHomology(int) 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.

Python

Not present. 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]

template<int dim>

MarkedAbelianGroup regina::detail::TriangulationBase< dim >::markedHomology ( int k ) const

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.

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.
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.

move20()

template<int dim>

template<int k>

bool regina::detail::TriangulationBase< dim >::move20 ( Face< dim, k > * f )

inlineinherited

If possible, performs a 2-0 move about the given k-face of degree two.

This involves taking the two top-dimensional simplices joined along that face and squashing them flat.

This move is currently only implemented for vertices, edges and triangles (i.e., facial dimension k ≤ 2).

This triangulation will be changed directly.

This move will only be performed if it will not change the topology of the manifold (as outlined below), and it will not violate any simplex and/or facet locks. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on locks.

The requirements for the move to not change the topology depend upon the facial dimension k. In all cases:

• the face f must be valid and non-boundary, and must have degree 2;
• the two top-dimensional simplices on either side of f must be distinct;
• the link of f must be the standard sphere obtained by identifying the boundaries of two (dim - k - 1)-simplices using the identity map;
• the two (dim - k - 1)-faces opposite f in each top-dimensional simplex must be distinct and not both boundary.

Moreover, there are further requirements depending on the facial dimension k:

• When performing the move on a vertex (k = 0), there are no additional requirements.
• When performing the move on an edge (k = 1), there are additional requirements on the (dim - 1)-faces. Specifically: the move would effectively flatten facets f1 and f2 of one top-dimensional simplex onto facets g1 and g2 of the other top-dimensional simplex respectiveyl, and we require that: (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 facets boundary and the other two identified.
• When performing the move on a triangle (k = 2), there are additional requirements on both the (dim - 1)-faces and the (dim - 2)-faces. These are move involved, and are discussed in detail in the source code for those who are interested.

If this triangulation is currently oriented, then this 2-0 move will preserve the orientation.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument f) can no longer be used.

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be 0, 1 or 2, and must not exceed dim - 2.

Parameters

f	the k-face about which to perform the move.

Returns

true if and only if the requested move was able to be performed.

moveContentsTo()

template<int dim>

void regina::detail::TriangulationBase< dim >::moveContentsTo ( Triangulation< dim > & dest )

inherited

Moves the contents of this triangulation into the given destination triangulation, leaving this triangulation empty but otherwise usable.

The top-dimensional simplices of this triangulation will be moved directly into dest, and placed after any pre-existing simplices. Specifically, if the original size of dest was N, then simplex(i) of this triangulation will become dest.simplex(N+i).

This triangulation will become empty as a result, but it will otherwise remain a valid and usable Triangulation object. Any simplex pointers that referred to either this triangulation or dest will remain valid (and will all now refer to dest), though if they originally referred to this triangulation then they will now return different indices. Any locks on top-dimensional simplices and/or their facets will be preserved.

Calling tri.moveContentsTo(dest) is similar to calling dest.insertTriangulation(std::move(tri)); it is a little slower but it comes with the benefit of leaving this triangulation in a usable state.

Precondition

dest is not this triangulation.

Parameters

dest	the triangulation into which the contents of this triangulation should be moved.

newSimplex() [1/2]

template<int dim>

Simplex< dim > * regina::detail::TriangulationBase< dim >::newSimplex ( )

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]

template<int dim>

Simplex< dim > * regina::detail::TriangulationBase< dim >::newSimplex ( const std::string & desc )

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.

newSimplexRaw()

template<int dim>

Simplex< dim > * regina::detail::TriangulationBase< dim >::newSimplexRaw ( )

inlineprotectedinherited

A variant of newSimplex() with no management of the underlying triangulation.

This routine adjusts the internal list of simplices, just like newSimplex() does. However, this is all it does. In particular:

• it does not manage the underlying triangulation in any way: it does not take snapshots, fire change events, or clear computed properties.

This should only be used in settings where the other missing tasks such as snapshots, change events and computed properties are being taken care of in some other manner (possibly manually). An example of such a setting might be the implementation of a local move (such as a Pachner move).

Such a "raw" routine would typically be safe to use without any manual error/lock/triangulation management in the following scenarios:

• triangulation constructors, but only in settings where no properties (including the skeleton) have been computed yet;
• routines that create a "staging" triangulation, without computing its skeleton or any other properties, and then swap or move this staging triangulation into the triangulation actually being worked upon (see subdivide() for an example).

The return value for this routine is the same as for newSimplex(). See newSimplex() for further details.

newSimplices() [1/2]

template<int dim>

template<int k>

std::array< Simplex< dim > *, k > regina::detail::TriangulationBase< dim >::newSimplices ( )

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:

auto [r, s, t] = ans.newSimplices<3>();
r->join(0, s, {1, 2, 3, 0});
...

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]

template<int dim>

void regina::detail::TriangulationBase< dim >::newSimplices ( size_t k )

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.

newSimplicesRaw()

template<int dim>

template<int k>

std::array< Simplex< dim > *, k > regina::detail::TriangulationBase< dim >::newSimplicesRaw ( )

inlineprotectedinherited

A variant of newSimplices() with no lock management, and no management of the underlying triangulation.

See newSimplexRaw() for further details on what these "raw" routines do and where they can be used.

The return value for this routine is the same as for newSimplices(). See newSimplices() for further details.

operator=() [1/2]

template<int dim>

Triangulation & regina::Triangulation< dim >::operator= ( const Triangulation< dim > & )

default

Sets this to be a (deep) copy of the given triangulation.

This will also clone any computed properties (such as homology, fundamental group, and so on), as well as the skeleton (vertices, edges, components, etc.). In particular, this triangulation will use the same numbering and labelling for all skeletal objects as in the source triangulation.

If src has any locks on top-dimensional simplices and/or their facets, these locks will also be copied across.

Returns

a reference to this triangulation.

operator=() [2/2]

template<int dim>

Triangulation & regina::Triangulation< dim >::operator= ( Triangulation< dim > && src )

default

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 top-dimensional simplex must be adjusted to point back to this triangulation instead of src.

All top-dimensional simplices 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 Simplex<dim>, Face<dim, subdim>, Component<dim> or BoundaryComponent<dim> objects will remain valid. Likewise, all cached properties will be moved into this triangulation.

If src has any locks on top-dimensional simplices and/or their facets, these locks will also be moved across.

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==()

template<int dim>

bool regina::detail::TriangulationBase< dim >::operator== ( const TriangulationBase< dim > & other ) const

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 or simplex/facet locks, as seen in Simplex<dim>::description() and Simplex<dim>::lockMask(); 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

true if and only if the two triangulations are combinatorially identical.

orient()

template<int dim>

void regina::detail::TriangulationBase< dim >::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 simplex 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.

If this triangulation has locks on any top-dimensional simplices and/or their facets, these will not prevent the orientation from taking place. Instead, any locks will be transformed accordingly (i.e., facets (dim - 1) and dim will exchange their lock states for those simplices that originally had negative orientation).

pachner() [1/3]

template<int dim>

template<int k>

bool regina::detail::TriangulationBase< dim >::pachner ( Face< dim, k > * f )

inlineinherited

If possible, 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 triangulation will be changed directly.

This move will only be performed if it will not change the topology of the manifold (as outlined below), and it will not violate any simplex and/or facet locks. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on locks.

In order to not change the topology, we require that:

• the given k-face is valid and non-boundary;
• the (dim + 1 - k) top-dimensional simplices that contain it are distinct; and
• 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 this triangulation is currently oriented, then this Pachner move will label the new top-dimensional simplices in a way that preserves the orientation.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be reconstructed, which means any pointers to old skeletal objects (such as the argument f) can no longer be used.

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

The check for this move takes quadratic time in the number of top-dimensional simplices involved (which may become expensive when dim is large and k is small). If you are certain that the move is legal, and you wish to circumvent this check, C++ users can call the variant of this function that takes an extra Unprotected argument.

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 now simplices().back()->vertex(0).

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be between 0 and (dim) inclusive.

Parameters

f	the k-face about which to perform the move.

Returns

true if and only if the requested move was able to be performed.

pachner() [2/3]

template<int dim>

template<int k>

bool regina::detail::TriangulationBase< dim >::pachner ( Face< dim, k > * f, bool ignored, bool perform = true )

inlineinherited

Deprecated routine that tests for and optionally performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face of this triangulation.

For more detail on Pachner moves and when they can be performed, see the variant of pachner() without the extra boolean arguments.

This routine will always check whether the requested move is legal and will not violate any simplex and/or facet locks (see Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on locks). If the move is allowed, and if the argument perform is true, this routine will also perform the move.

Deprecated

If you just wish to test whether such a move is possible, call hasPachner(). If you wish to both check and perform the move, call pachner() without the two extra boolean arguments.

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be between 0 and (dim) inclusive.

Parameters

f	the k-face about which to perform the move.
ignored	an argument that is ignored. In earlier versions of Regina this argument controlled whether we check if the move can be performed; however, now this check is done always.
perform	true if we should actually perform the move, assuming the move is allowed.

Returns

true if and only if the requested move could be performed.

pachner() [3/3]

template<int dim>

template<int k>

void regina::detail::TriangulationBase< dim >::pachner ( Face< dim, k > * f, Unprotected  )

inlineinherited

Performs a (dim + 1 - k)-(k + 1) Pachner move about the given k-face, without any safety checks.

This variant of pachner() is offered because it can become expensive to test whether a Pachner move can be performed on a given face when dim is large and k is small (and therefore there are many top-dimensional simplices involved in the move).

This function will always perform the requested Pachner move directy on this triangulation, without first testing whether it is legal. The onus is on the programmer to ensure the legality of the move beforehand. Getting this wrong could be disastrous: you could change the topology of the manifold, or possibly end up with an invalid triangulation, or this function could throw an exception if you attempt to violate a simplex and/or facet lock.

The (unnamed) Unprotected argument would typically be the constant regina::unprotected.

For more detail on Pachner moves and when they can be performed, see the variant of pachner() that does not take an extra Unprotected argument.

Precondition

The given k-face is a k-face of this triangulation.

Python

Not present. By design, Python users are not able to circumvent the legality checks for Pachner moves. If speed is essential, you should be using C++.

Exceptions

LockViolation

This move would violate a simplex or facet lock. This exception will be thrown before any changes are made.

Template Parameters

k	the dimension of the given face. This must be between 0 and (dim) inclusive.

Parameters

f	the k-face about which to perform the move.

packet() [1/2]

std::shared_ptr< PacketOf< Triangulation< dim > > > regina::PacketData< Triangulation< dim > >::packet ( )

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 null if this data is not (directly) held by a packet.

packet() [2/2]

std::shared_ptr< const PacketOf< Triangulation< dim > > > regina::PacketData< Triangulation< dim > >::packet ( ) const

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 null if this data is not (directly) held by a packet.

pairing()

template<int dim>

FacetPairing< dim > regina::detail::TriangulationBase< dim >::pairing ( ) const

inlineinherited

Returns the dual graph of this triangulation, expressed as a facet pairing.

Calling tri.pairing() is equivalent to calling FacetPairing<dim>(tri).

Precondition

This triangulation is not empty.

Returns

the dual graph of this triangulation.

pentachora()

template<int dim>

auto regina::detail::TriangulationBase< dim >::pentachora ( ) const

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]

template<int dim>

auto regina::detail::TriangulationBase< dim >::pentachoron ( size_t index )

inlineinherited

A dimension-specific alias for face<4>(), or an alias for simplex() in dimension dim = 4.

This alias is available for dimensions dim ≥ 4. It returns a non-const pentachoron pointer.

See face() for further information.

pentachoron() [2/2]

template<int dim>

auto regina::detail::TriangulationBase< dim >::pentachoron ( size_t index ) const

inlineinherited

A dimension-specific alias for face<4>(), or an alias for simplex() in dimension dim = 4.

This alias is available for dimensions dim ≥ 4. It returns a const pentachoron pointer in dimension dim = 4, and a non-const pentachoron pointer in all higher dimensions.

See face() for further information.

randomiseLabelling()

template<int dim>

Isomorphism< dim > regina::detail::TriangulationBase< dim >::randomiseLabelling ( bool preserveOrientation = true )

inlineinherited

Randomly relabels the top-dimensional simplices and their vertices.

Essentially, this routine creates a random isomorphism of the correct size and applies it in-place to this triangulation.

The advantage of using this routine instead of working directly through the Isomorphism class is that this routine preserves any computed topological properties of the triangulation (as opposed to the isomorphism bracket operator, which at the time of writing does not).

Parameters

preserveOrientation

if true, then every top-dimensional simplex will have its vertices permuted with an even permutation. This means that, if this triangulation is oriented, then randomiseLabelling() will preserve the orientation.

Returns

the random isomorphism that was applied; that is, the isomorphism from the original triangulation to the final triangulation.

reflect()

template<int dim>

void regina::detail::TriangulationBase< dim >::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.

If this triangulation has locks on any top-dimensional simplices and/or their facets, these will not prevent the reflection from taking place. Instead, any locks will be transformed accordingly (i.e., facets (dim - 1) and dim will exchange their lock states in every top-dimensional simplex).

removeAllSimplices()

template<int dim>

void regina::detail::TriangulationBase< dim >::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.

Exceptions

LockViolation

This triangulation contains at least one locked top-dimensional simplex and/or facet. This exception will be thrown before any changes are made. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on how such locks work and what their implications are.

removeSimplex()

template<int dim>

void regina::detail::TriangulationBase< dim >::removeSimplex ( Simplex< dim > * simplex )

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.

Exceptions

LockViolation

The given simplex and/or one of its facets is currently locked. This exception will be thrown before any changes are made. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on how such locks work and what their implications are.

Parameters

simplex

the simplex to remove.

removeSimplexAt()

template<int dim>

void regina::detail::TriangulationBase< dim >::removeSimplexAt ( size_t index )

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.

Exceptions

LockViolation

The requested simplex and/or one of its facets is currently locked. This exception will be thrown before any changes are made. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on how such locks work and what their implications are.

Parameters

index

specifies which top-dimensional simplex to remove; this must be between 0 and size()-1 inclusive.

removeSimplexRaw()

template<int dim>

void regina::detail::TriangulationBase< dim >::removeSimplexRaw ( Simplex< dim > * simplex )

inlineprotectedinherited

A variant of removeSimplex() with no lock management, and no management of the underlying triangulation.

See newSimplexRaw() for further details on what these "raw" routines do and where they can be used.

The arguments for this routine are the same as for removeSimplex(). See removeSimplex() for further details.

reorderBFS()

template<int dim>

void regina::detail::TriangulationBase< dim >::reorderBFS ( bool reverse = false )

inherited

Reorders the top-dimensional simplices of this triangulation using a breadth-first search, so that small-numbered simplices are adjacent to other small-numbered simplices.

Specifically, the reordering will operate as follows. Simplex 0 will remain simplex 0. Its immediate neighbours will be numbered 1,2,...,(dim+1) (though if these neighbours are not distinct then of course fewer labels will be required). Their immediate neighbours will in turn be numbered (dim+2), (dim+3) and so on, ultimately following a breadth-first search throughout the entire triangulation.

If the optional argument reverse is true, then simplex numbers will be assigned in reverse order. That is, simplex 0 will become simplex n-1, its immediate neighbours will become simplices n-2, n-3, etc., and so on.

If this triangulation has locks on any top-dimensional simplices and/or their facets, these will not prevent the reordering from taking place. Instead, any locks will be transformed accordingly; that is, all top-dimensional simplices will carry their own locks and their facets' locks around with them as they are reordered.

Parameters

reverse

true if the new simplex numbers should be assigned in reverse order, as described above.

setGroupPresentation()

template<int dim>

void regina::detail::TriangulationBase< dim >::setGroupPresentation ( GroupPresentation pres )

inlineinherited

Allows the specific presentation of the fundamental group to be changed by some other (external) means.

Specifically, this routine assumes that you have changed (and presumably simplified) the presentation of the fundamental group using some external tool (such as GAP or Magma), and it replaces the current presentation with the new presentation pres that is passed.

Regina does not verify that the new presentation is isomorphic to the old, since this is an extremely hard problem.

If the fundamental group has not yet been calculated for this triangulation, then this routine will cache pres 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 group() at all.

Note that this routine will not fire a packet change event.

Precondition

The given presentation pres is indeed a presentation of the fundamental group of this triangulation, as described by group().

Parameters

pres	a new presentation of the fundamental group of this triangulation.

shellBoundary() [1/2]

template<int dim>

bool regina::detail::TriangulationBase< dim >::shellBoundary ( Simplex< dim > * s )

inlineinherited

If possible, performs a boundary shelling move upon the given top-dimensional simplex of this triangulation.

This involves popping off a top-dimensional simplex with one or more facets on the boundary.

This move is only available in standard dimensions, since Regina's notion of "valid faces" is weaker in higher dimensions (due to the need to solve undecidable problems). See Face::isValid() for further discussion.

This triangulation will be changed directly.

This move will only be performed if it will not change the topology of the manifold (as outlined below), and it will not violate any simplex and/or facet locks. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on locks.

In order to not change the topology, we require the conditions below. These are conditions are stricter than necessary, but the resulting "lost opportunities" only affect invalid triangulations.

In this list of conditions, we let s denote the given top-dimensional simplex, and we let f denote the face of s opposite the intersection of all its boundary facets. For example, for a 4-manifold triangulation where s has exactly two boundary facets meeting in a common boundary triangle, f would denote the edge of s opposite this common triangle. Put differently: f is the (unique) smallest face of s that might not be entirely contained in the triangulation boundary. Note that the dimension of f will be one less than the number of boundary facets of s.

Having said all of this, our conditions are:

• all faces of all dimensions of the simplex s, except possibly its vertices, are valid;
• at least one but not all of the facets of s lie in the boundary of the triangulation;
• the face f (defined above) is valid (even if it is a vertex), and does not lie entirely in the boundary of the triangulation;
• for each facial dimension k, no two k-faces of s that both contain f are identified (testing this for k = dim(f)+1 is enough to ensure it holds for all k).

If this triangulation is currently oriented, then this operation will (trivially) preserve the orientation.

Note that after performing this move, all skeletal objects (faces, components, etc.) will be reconstructed, which means any pointers to old skeletal objects can no longer be used.

Precondition

The dimension dim is one of Regina's standard dimensions.

The given simplex is a simplex of this triangulation.

Parameters

s	the top-dimensional simplex upon which to perform the move.

Returns

true if and only if the requested move was able to be performed.

shellBoundary() [2/2]

template<int dim>

bool regina::detail::TriangulationBase< dim >::shellBoundary ( Simplex< dim > * s, bool ignored, bool perform = true )

inlineinherited

Deprecated routine that tests for and optionally performs a boundary shelling move on the given top-dimensional simplex.

For more detail on boundary shelling moves and when they can be performed, see the variant of shellBoundary() without the extra boolean arguments.

This routine will always check whether the requested move is legal and will not violate any simplex and/or facet locks (see Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on locks). If the move is allowed, and if the argument perform is true, this routine will also perform the move.

This move is only available in standard dimensions, since Regina's notion of "valid faces" is weaker in higher dimensions (due to the need to solve undecidable problems). See Face::isValid() for further discussion.

Deprecated

If you just wish to test whether such a move is possible, call hasShellBoundary(). If you wish to both check and perform the move, call shellBoundary() without the two extra boolean arguments.

Precondition

The dimension dim is one of Regina's standard dimensions.

The given simplex is a simplex of this triangulation.

Parameters

s	the top-dimensional simplex upon which to perform the move.
ignored	an argument that is ignored. In earlier versions of Regina this argument controlled whether we check if the move can be performed; however, now this check is done always.
perform	true if we should actually perform the move, assuming the move is allowed.

Returns

true if and only if the requested move could be performed.

sig()

template<int dim>

template<class Type , class Encoding >

Encoding::Signature regina::detail::TriangulationBase< dim >::sig ( ) const

inlineinherited

Alias for isoSig(), which constructs the isomorphism signature of the given type for this triangulation.

This alias sig() is provided to assist with generic code that can work with both triangulations and links.

See isoSig() for further details.

Python

This alias is only available for the default signature type and encoding (i.e., the default C++ template arguments). If you wish to use a different signature type and/or encoding, you can instead use the variants provided with isoSig(); that is, you can call a function of the form isoSig_Type. See the isoSig() documentation for further details.

Returns

the isomorphism signature of this triangulation.

simplex() [1/2]

template<int dim>

Simplex< dim > * regina::detail::TriangulationBase< dim >::simplex ( size_t index )

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]

template<int dim>

const Simplex< dim > * regina::detail::TriangulationBase< dim >::simplex ( size_t index ) const

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()

template<int dim>

auto regina::detail::TriangulationBase< dim >::simplices ( ) const

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 range-based for loops. Note that the elements of the list will be pointers, so your code might look like:

for (Simplex<dim>* s : tri.simplices()) { ... }

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()

template<int dim>

void regina::detail::TriangulationBase< dim >::simplifiedFundamentalGroup ( GroupPresentation pres )

inlineinherited

Deprecated alias for setGroupPresentation(), which allows the specific presentation of the fundamental group to be changed by some other (external) means.

Deprecated

This routine has been renamed to setGroupPresentation().

Precondition

The given presentation pres is indeed a presentation of the fundamental group of this triangulation, as described by group().

Parameters

pres	a new presentation of the fundamental group of this triangulation.

size()

template<int dim>

size_t regina::detail::TriangulationBase< dim >::size ( ) const

inlineinherited

Returns the number of top-dimensional simplices in the triangulation.

Returns

The number of top-dimensional simplices.

source()

template<int dim>

std::string regina::detail::TriangulationBase< dim >::source ( Language language = Language::Current ) const

inherited

Returns C++ or Python source code that can be used to reconstruct this triangulation.

This code will call Triangulation<dim>::fromGluings(), passing a hard-coded C++ initialiser list or Python list (depending on the requested language).

The main purpose of this routine is to generate this hard-coded 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.

Parameters

language

the language in which the source code should be written.

Returns

the source code that was generated.

str()

std::string regina::Output< Triangulation< dim >, false >::str ( ) const

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 Python __repr__() function will incorporate this into its output.

Returns

a short text representation of this object.

subdivide()

template<int dim>

void regina::detail::TriangulationBase< dim >::subdivide ( )

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).

Exceptions

LockViolation

This triangulation contains at least one locked top-dimensional simplex and/or facet. This exception will be thrown before any changes are made. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on how such locks work and what their implications are.

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).

swap() [1/2]

void regina::Snapshottable< Triangulation< dim > >::swap ( Snapshottable< Triangulation< dim > > & other )

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]

template<int dim>

void regina::Triangulation< dim >::swap

(

Triangulation< dim > &

other

)

Swaps the contents of this and the given triangulation.

All top-dimensional simplices that belong to this triangulation will be moved to other, and all top-dimensional simplices 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 will be swapped.

In particular, any pointers or references to Simplex<dim> and/or Face<dim, 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()

template<int dim>

void regina::detail::TriangulationBase< dim >::swapBaseData ( TriangulationBase< dim > & other )

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.

takeSnapshot()

void regina::Snapshottable< Triangulation< dim > >::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()

template<int dim>

auto regina::detail::TriangulationBase< dim >::tetrahedra ( ) const

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]

template<int dim>

auto regina::detail::TriangulationBase< dim >::tetrahedron ( size_t index )

inlineinherited

A dimension-specific alias for face<3>(), or an alias for simplex() in dimension dim = 3.

This alias is available for dimensions dim ≥ 3. It returns a non-const tetrahedron pointer.

See face() for further information.

tetrahedron() [2/2]

template<int dim>

auto regina::detail::TriangulationBase< dim >::tetrahedron ( size_t index ) const

inlineinherited

A dimension-specific alias for face<3>(), or an alias for simplex() in dimension dim = 3.

This alias is available for dimensions dim ≥ 3. It returns a const tetrahedron pointer in dimension dim = 3, and a non-const tetrahedron pointer in all higher dimensions.

See face() for further information.

tightDecode()

template<int dim>

Triangulation< dim > regina::detail::TriangulationBase< dim >::tightDecode ( std::istream & input )

staticinherited

Reconstructs a triangulation from its given tight encoding.

See the page on tight encodings for details.

The tight encoding will be read from the given input stream. If the input stream contains leading whitespace then it will be treated as an invalid encoding (i.e., this routine will throw an exception). The input stream may contain further data: if this routine is successful then the input stream will be left positioned immediately after the encoding, without skipping any trailing whitespace.

Exceptions

InvalidInput

The given input stream does not begin with a tight encoding of a dim-dimensional triangulation.

Python

Not present. Use tightDecoding() instead, which takes a string as its argument.

Parameters

input

an input stream that begins with the tight encoding for a dim-dimensional triangulation.

Returns

the triangulation represented by the given tight encoding.

tightDecoding()

static Triangulation< dim > regina::TightEncodable< Triangulation< dim > >::tightDecoding ( const std::string & enc )

inlinestaticinherited

Reconstructs an object of type T from its given tight encoding.

See the page on tight encodings for details.

The tight encoding should be given as a string. If this string contains leading whitespace or any trailing characters at all (including trailing whitespace), then it will be treated as an invalid encoding (i.e., this routine will throw an exception).

Exceptions

InvalidArgument

The given string is not a tight encoding of an object of type T.

Parameters

enc	the tight encoding for an object of type T.

Returns

the object represented by the given tight encoding.

tightEncode()

template<int dim>

void regina::detail::TriangulationBase< dim >::tightEncode ( std::ostream & out ) const

inherited

Writes the tight encoding of this triangulation to the given output stream.

See the page on tight encodings for details.

Python

Not present. Use tightEncoding() instead, which returns a string.

Parameters

out	the output stream to which the encoded string will be written.

tightEncoding()

std::string regina::TightEncodable< Triangulation< dim > >::tightEncoding ( ) const

inlineinherited

Returns the tight encoding of this object.

See the page on tight encodings for details.

Exceptions

FailedPrecondition

This may be thrown for some classes T if the object is in an invalid state. If this is possible, then a more detailed explanation of "invalid" can be found in the class documentation for T, under the member function T::tightEncode(). See FacetPairing::tightEncode() for an example of this.

Returns

the resulting encoded string.

topologyLocked()

bool regina::TopologyLockable::topologyLocked ( ) const

inlineprotectedinherited

Returns whether or not there are any topology locks currently held on this object.

Strictly speaking, this routine could return a false negative: the number of locks is stored as an 8-bit integer and so in reality this tests whether the number of locks is a multiple of 256. False negatives are mathematically harmless, since the worst that will happen is that topological properties will be cleared when they could have been preserved, and so unnecessary extra computation may be required to compute them again.

This routine will never return a false positive.

Returns

false if there are no topology locks currently held on this object, or if a false negative occurs (as described above); or true to indicate that there are currently topology locks held on this object.

translate() [1/2]

template<int dim>

template<int subdim>

Face< dim, subdim > * regina::detail::TriangulationBase< dim >::translate ( const Face< dim, subdim > * other ) const

inlineinherited

Translates a face of some other triangulation into the corresponding face of this triangulation, using simplex numbers for the translation.

Typically this routine would be used when the given face comes from a triangulation that is combinatorially identical to this, and you wish to obtain the corresponding face of this triangulation.

Specifically:

• For faces of dimension k < dim, if other refers to face i of top-dimensional simplex number k of some other triangulation, then this routine will return face i of top-dimensional simplex number k of this triangulation. Note that this routine does not use the face indices within each triangulation (which is outside the user's control), but rather the simplex numbering (which the user has full control over).
• For top-dimensional simplices (i.e., faces of dimension k = dim), this routine will simply translate top-dimensional simplex number k of some other triangulation into top-dimensional simplex number k of this triangulation.

This routine behaves correctly even if other is a null pointer.

Precondition

This triangulation contains at least as many top-dimensional simplices as the triangulation containing other (though, as noted above, in typical scenarios both triangulations would actually be combinatorially identical).

Template Parameters

subdim

the face dimension; this must be between 0 and dim inclusive.

Parameters

other

the face to translate.

Returns

the corresponding face of this triangulation.

translate() [2/2]

template<int dim>

template<int subdim>

FaceEmbedding< dim, subdim > regina::detail::TriangulationBase< dim >::translate ( const FaceEmbedding< dim, subdim > & other ) const

inlineinherited

Translates a face embedding from some other triangulation into the corresponding face embedding with respect to this triangulation, using simplex numbers for the translation.

Typically this routine would be used when the given embedding comes from a triangulation that is combinatorially identical to this, and you wish to obtain the corresponding face embedding within this triangulation.

Specifically: if the embedding other refers to face i of top-dimensional simplex number k of some other triangulation, then the embedding that is returned will refer to face i of top-dimensional simplex number k of this triangulation.

Precondition

This triangulation contains at least as many top-dimensional simplices as the triangulation containing other (though, as noted above, in typical scenarios both triangulations would actually be combinatorially identical).

Template Parameters

subdim

the face dimension; this must be between 0 and dim-1 inclusive.

Parameters

other

the face embedding to translate.

Returns

the corresponding face embedding in this triangulation.

triangle() [1/2]

template<int dim>

auto regina::detail::TriangulationBase< dim >::triangle ( size_t index )

inlineinherited

A dimension-specific alias for face<2>(), or an alias for simplex() in dimension dim = 2.

This alias is available for all dimensions dim. It returns a non-const triangle pointer.

See face() for further information.

triangle() [2/2]

template<int dim>

auto regina::detail::TriangulationBase< dim >::triangle ( size_t index ) const

inlineinherited

A dimension-specific alias for face<2>(), or an alias for simplex() in dimension dim = 2.

This alias is available for all dimensions dim. It returns a const triangle pointer in dimension dim = 2, and a non-const triangle pointer in all higher dimensions.

See face() for further information.

triangles()

template<int dim>

auto regina::detail::TriangulationBase< dim >::triangles ( ) const

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()

template<int dim>

std::vector< Triangulation< dim > > regina::detail::TriangulationBase< dim >::triangulateComponents ( ) const

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.

If this triangulation has locks on any top-dimensional simplices and/or their facets, these locks will also be copied over to the newly-triangulated components.

Returns

a list of individual component triangulations.

twoZeroMove()

template<int dim>

template<int k>

bool regina::detail::TriangulationBase< dim >::twoZeroMove ( Face< dim, k > * f, bool ignored, bool perform = true )

inlineinherited

Deprecated routine that tests for and optionally performs a 2-0 move about the given k-face of this triangulation.

For more detail on 2-0 moves and when they can be performed, see move20().

This routine will always check whether the requested move is legal and will not violate any simplex and/or facet locks (see Simplex<dim>::lock() and Simplex<dim>::lockFacet() for further details on locks). If the move is allowed, and if the argument perform is true, this routine will also perform the move.

Deprecated

If you just wish to test whether such a move is possible, call has20(). If you wish to both check and perform the move, call move20(), which does not take the two extra boolean arguments.

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be 0, 1 or 2, and must not exceed dim - 2.

Parameters

f	the k-face about which to perform the move.
ignored	an argument that is ignored. In earlier versions of Regina this argument controlled whether we check if the move can be performed; however, now this check is done always.
perform	true if we should actually perform the move, assuming the move is allowed.

Returns

true if and only if the requested move could be performed.

unlockAll()

template<int dim>

void regina::detail::TriangulationBase< dim >::unlockAll ( )

inherited

Unlocks all top-dimensional simplices and their facets.

In short, locking a top-dimensional simplex and/or some of its facets means that that the simplex and/or facets must not be changed. See Simplex<dim>::lock() and Simplex<dim>::lockFacet() for full details on how locks work and what their implications are.

After this is routine called, hasLocks() will return false.

utf8()

std::string regina::Output< Triangulation< dim >, false >::utf8 ( ) const

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()

template<int dim>

Face< dim, 0 > * regina::detail::TriangulationBase< dim >::vertex ( size_t index ) const

inlineinherited

A dimension-specific alias for face<0>().

This alias is available for all dimensions dim.

See face() for further information.

vertices()

template<int dim>

auto regina::detail::TriangulationBase< dim >::vertices ( ) const

inlineinherited

A dimension-specific alias for faces<0>().

This alias is available for all dimensions dim.

See faces() for further information.

with20()

template<int dim>

template<int k>

std::optional< Triangulation< dim > > regina::detail::TriangulationBase< dim >::with20 ( Face< dim, k > * f ) const

inherited

If possible, returns the triangulation obtained by performing a 2-0 move about the given k-face of this triangulation.

If such a move is not allowed, or if such a move would violate any simplex and/or facet locks, then this routine returns no value.

This triangulation will not be changed.

For more detail on 2-0 moves and when they can be performed, see move20().

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be 0, 1 or 2, and must not exceed dim - 2.

Parameters

f	the k-face about which to perform the move.

Returns

The new triangulation obtained by performing the requested move, or no value if the requested move cannot be performed.

withPachner()

template<int dim>

template<int k>

std::optional< Triangulation< dim > > regina::detail::TriangulationBase< dim >::withPachner ( Face< dim, k > * f ) const

inherited

If possible, returns the triangulation obtained by performing a (dim + 1 - k)-(k + 1) Pachner move about the given k-face of this triangulation.

If such a move is not allowed, or if such a move would violate any simplex and/or facet locks, then this routine returns no value.

This triangulation will not be changed.

For more detail on Pachner moves and when they can be performed, see pachner().

Precondition

The given k-face is a k-face of this triangulation.

Template Parameters

k	the dimension of the given face. This must be between 0 and (dim) inclusive.

Parameters

f	the k-face about which to perform the move.

Returns

The new triangulation obtained by performing the requested move, or no value if the requested move cannot be performed.

withShellBoundary()

template<int dim>

std::optional< Triangulation< dim > > regina::detail::TriangulationBase< dim >::withShellBoundary ( Simplex< dim > * s ) const

inherited

If possible, returns the triangulation obtained by performing a boundary shelling move on the given top-dimensional simplex of this triangulation.

If such a move is not allowed, or if such a move would violate any simplex and/or facet locks, then this routine returns no value.

This operation is only available in standard dimensions, since Regina's notion of "valid faces" is weaker in higher dimensions (due to the need to solve undecidable problems). See Face::isValid() for further discussion.

This triangulation will not be changed.

For more detail on boundary shelling moves and when they can be performed, see shellBoundary().

Precondition

The dimension dim is one of Regina's standard dimensions.

The given simplex is a simplex of this triangulation.

Parameters

s	the top-dimensional simplex upon which to perform the move.

Returns

The new triangulation obtained by performing the requested move, or no value if the requested move cannot be performed.

writeDot()

template<int dim>

void regina::detail::TriangulationBase< dim >::writeDot ( std::ostream & out, bool labels = false ) const

inherited

Writes the dual graph of this triangulation in the Graphviz DOT language.

See dot() for further details on what this output contains.

This routine is equivalent to (but faster than) writing the string returned by dot() to the given output stream.

Python

Not present. Use dot() instead, which returns a string.

Parameters

out	the output stream to which to write.
labels	indicates whether graph vertices should be labelled with the corresponding top-dimensional simplex numbers.

See also

http://www.graphviz.org/

writeTextLong()

template<int dim>

void regina::detail::TriangulationBase< dim >::writeTextLong ( std::ostream & out ) const

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()

template<int dim>

void regina::detail::TriangulationBase< dim >::writeTextShort ( std::ostream & out ) const

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()

template<int dim>

void regina::detail::TriangulationBase< dim >::writeXMLBaseProperties ( std::ostream & out ) const

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.

Member Data Documentation

boundaryComponents_

template<int dim>

MarkedVector<BoundaryComponent<dim> > regina::detail::TriangulationBase< dim >::boundaryComponents_

protectedinherited

The components that form the boundary of the triangulation.

components_

template<int dim>

MarkedVector<Component<dim> > regina::detail::TriangulationBase< dim >::components_

protectedinherited

The connected components that form the triangulation.

This list is only filled if/when the skeleton of the triangulation is computed.

dimension

template<int dim>

int regina::detail::TriangulationBase< dim >::dimension = dim

staticconstexprinherited

A compile-time constant that gives the dimension of the triangulation.

heldBy_

PacketHeldBy regina::PacketData< Triangulation< dim > >::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.

nBoundaryFaces_

template<int dim>

std::array<size_t, dim> regina::detail::TriangulationBase< dim >::nBoundaryFaces_

protectedinherited

The number of boundary faces of each dimension.

simplices_

template<int dim>

MarkedVector<Simplex<dim> > regina::detail::TriangulationBase< dim >::simplices_

protectedinherited

The top-dimensional simplices that form the triangulation.

topologyLock_

uint8_t regina::TopologyLockable::topologyLock_ { 0 }

protectedinherited

The number of topology locks currently held on this object.

Any non-zero number of locks implies that "hook routines" that clear computed properties (as described in the class notes) will preserve properties that are purely topological.

valid_

template<int dim>

bool regina::detail::TriangulationBase< dim >::valid_

protectedinherited

Is this triangulation valid? See isValid() for details on what this means.

---

The documentation for this class was generated from the following files:

• link/link.h
• triangulation/cut.h
• triangulation/forward.h
• triangulation/generic/triangulation.h