Represents a three-tetrahedron triangular solid torus in a triangulation. More...

#include <subcomplex/trisolidtorus.h>

Inheritance diagram for regina::TriSolidTorus:

Public Member Functions
	TriSolidTorus (const TriSolidTorus &)=default
	Creates a new copy of the given structure. More...

TriSolidTorus &	operator= (const TriSolidTorus &)=default
	Sets this to be a copy of the given structure. More...

void	swap (TriSolidTorus &other) noexcept
	Swaps the contents of this and the given structure. More...

Tetrahedron< 3 > *	tetrahedron (int index) const
	Returns the requested tetrahedron in this solid torus. More...

Perm< 4 >	vertexRoles (int index) const
	Returns a permutation represeting the role that each vertex of the requested tetrahedron plays in the solid torus. More...

std::optional< Perm< 4 > >	isAnnulusSelfIdentified (int index) const
	Determines whether the two triangles of the requested annulus are glued to each other. More...

size_t	areAnnuliLinkedMajor (int otherAnnulus) const
	Determines whether the two given annuli are linked in a particular fashion by a layered chain. More...

size_t	areAnnuliLinkedAxis (int otherAnnulus) const
	Determines whether the two given annuli are linked in a particular fashion by a layered chain. More...

bool	operator== (const TriSolidTorus &other) const
	Determines whether this and the given object represent the same specific presentation of a triangular solid torus. More...

bool	operator!= (const TriSolidTorus &other) const
	Determines whether this and the given object represent different specific presentations of a triangular solid torus. More...

std::unique_ptr< Manifold >	manifold () const override
	Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented. More...

AbelianGroup	homology () const override
	Returns the expected first homology group of this triangulation, if such a routine has been implemented. More...

std::ostream &	writeName (std::ostream &out) const override
	Writes the name of this triangulation as a human-readable string to the given output stream. More...

std::ostream &	writeTeXName (std::ostream &out) const override
	Writes the name of this triangulation in TeX format to the given output stream. More...

void	writeTextShort (std::ostream &out) const override
	Writes a short text representation of this object to the given output stream. More...

std::string	name () const
	Returns the name of this specific triangulation as a human-readable string. More...

std::string	texName () const
	Returns the name of this specific triangulation in TeX format. More...

virtual void	writeTextLong (std::ostream &out) const
	Writes a detailed text representation of this object to the given output stream. More...

std::string	str () const
	Returns a short text representation of this object. More...

std::string	utf8 () const
	Returns a short text representation of this object using unicode characters. More...

std::string	detail () const
	Returns a detailed text representation of this object. More...

Static Public Member Functions
static std::unique_ptr< TriSolidTorus >	recognise (Tetrahedron< 3 > *tet, Perm< 4 > useVertexRoles)
	Determines if the given tetrahedron forms part of a three-tetrahedron triangular solid torus with its vertices playing the given roles in the solid torus. More...

static std::unique_ptr< StandardTriangulation >	recognise (Component< 3 > *component)
	Determines whether the given component represents one of the standard triangulations understood by Regina. More...

static std::unique_ptr< StandardTriangulation >	recognise (const Triangulation< 3 > &tri)
	Determines whether the given triangulation represents one of the standard triangulations understood by Regina. More...

Detailed Description

Represents a three-tetrahedron triangular solid torus in a triangulation.

A three-tetrahedron triangular solid torus is a three-tetrahedron triangular prism with its two ends identified.

The resulting triangular solid torus will have all edges as boundary edges. Three of these will be axis edges (parallel to the axis of the solid torus). Between the axis edges will be three annuli, each with two internal edges. One of these internal edges will meet all three tetrahedra (the major edge) and one of these internal edges will only meet two of the tetrahedra (the minor edge).

Assume the axis of the layered solid torus is oriented. The three major edges together form a loop on the boundary torus. This loop can be oriented to run around the solid torus in the same direction as the axis; this then induces an orientation on the boundary of a meridinal disc. Thus, using an axis edge as longitude, the three major edges will together form a (1,1) curve on the boundary torus.

We can now orient the minor edges so they also run around the solid torus in the same direction as the axis, together forming a (2, -1) curve on the boundary torus.

Finally, the three tetrahedra can be numbered 0, 1 and 2 in an order that follows the axis, and the annuli can be numbered 0, 1 and 2 in an order that follows the meridinal disc boundary so that annulus i does not use any faces from tetrahedron i.

Note that all three tetrahedra in the triangular solid torus must be distinct.

All optional StandardTriangulation routines are implemented for this class.

This class supports copying but does not implement separate move operations, since its internal data is so small that copying is just as efficient. It implements the C++ Swappable requirement via its own member and global swap() functions, for consistency with the other StandardTriangulation subclasses. Note that the only way to create these objects (aside from copying or moving) is via the static member function recognise().

Constructor & Destructor Documentation

◆ TriSolidTorus()

regina::TriSolidTorus::TriSolidTorus ( const TriSolidTorus & )

default

Creates a new copy of the given structure.

Member Function Documentation

◆ areAnnuliLinkedAxis()

size_t regina::TriSolidTorus::areAnnuliLinkedAxis ( int otherAnnulus ) const

Determines whether the two given annuli are linked in a particular fashion by a layered chain.

In this scenario, one of the given annuli meets both faces of the top tetrahedron and the other annulus meets both faces of the bottom tetrahedron of the layered chain.

To be identified by this routine, the layered chain (described by LayeredChain) must be attached as follows.

The two diagonals of the layered chain (between the two top faces and between the two bottom faces) should correspond to the two directed major edges of the two annuli, with the major edges both pointing from top hinge edge to bottom hinge edge. The other boundary edges of the layered chain that are not hinge edges should correspond to the two directed minor edges of the two annuli, with the minor edges both pointing from bottom hinge edge to top hinge edge. The hinge edges themselves should correspond to the axis edges of the triangular solid torus (this correspondence is determined by the previous identifications; the axis edge between the two annuli will be identified to both of the others in reverse).

Parameters

otherAnnulus the annulus on the solid torus boundary not to be examined; this must be 0, 1 or 2.

Returns: the number of tetrahedra in the layered chain if the two annuli are linked as described, or 0 otherwise.

◆ areAnnuliLinkedMajor()

size_t regina::TriSolidTorus::areAnnuliLinkedMajor ( int otherAnnulus ) const

Determines whether the two given annuli are linked in a particular fashion by a layered chain.

In this scenario, both of the given annuli meet one face of the top tetrahedron and one face of the bottom tetrahedron of the layered chain.

To be identified by this routine, the layered chain (described by LayeredChain) must be attached as follows. The two directed major edges of the two annuli should correspond to the two hinge edges of the layered chain (with both hinge edges pointing in the same direction around the solid torus formed by the layered chain). The two directed diagonals of the layered chain (between the two top faces and between the two bottom faces, each pointing in the opposite direction to the hinge edges around the solid torus formed by the layered chain) should be identified and must correspond to the (identified) two directed minor edges of the two annuli. The remaining boundary edges of the layered chain should correspond to the axis edges of the triangular solid torus (this correspondence is determined by the previous identifications).

Parameters

otherAnnulus the annulus on the solid torus boundary not to be examined; this must be 0, 1 or 2.

Returns: the number of tetrahedra in the layered chain if the two annuli are linked as described, or 0 otherwise.

◆ detail()

std::string regina::Output< StandardTriangulation , 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.

◆ homology()

AbelianGroup regina::TriSolidTorus::homology ( ) const

overridevirtual

Returns the expected first homology group of this triangulation, if such a routine has been implemented.

This routine does not work by calling Triangulation<3>::homology() on the associated real triangulation. Instead the homology is calculated directly from the known properties of this standard triangulation.

This means that homology() needs to be implemented separately for each class of standard triangulation. See the class notes for each subclass of StandardTriangulation for details on whether homology has been implemented for that particular subclass. The default implementation of this routine just throws a NotImplemented exception.

Most users will not need this routine, since presumably you already have an explicit Triangulation<3> available and so you can just call Triangulation<3>::homology() instead (which, unlike this routine, is always implemented). This StandardTriangulation::homology() routine should be seen as more of a verification/validation tool for the Regina developers.

If this StandardTriangulation describes an entire Triangulation<3> (and not just a part thereof) then the results of this routine should be identical to the homology group obtained by calling Triangulation<3>::homology() upon the associated real triangulation.

Exceptions

NotImplemented	Homology calculation has not yet been implemented for this particular type of standard triangulation.
FileError	The homology needs to be read from file (as opposed to computed), but the file is inaccessible or its contents cannot be read and parsed correctly. Currently this can only happen for the subclass SnapPeaCensusTri, which reads its results from the SnapPea census databases that are installed with Regina.

Returns: the first homology group of this triangulation, if this functionality has been implemented.

Reimplemented from regina::StandardTriangulation.

◆ isAnnulusSelfIdentified()

std::optional< Perm< 4 > > regina::TriSolidTorus::isAnnulusSelfIdentified ( int index ) const

Determines whether the two triangles of the requested annulus are glued to each other.

If the two triangles are glued, this routine will return a permutation describing how the vertex roles are glued to each other. This will describe directly how axis edges, major edges and minor edges map to each other without having to worry about the specific assignment of tetrahedron vertex numbers. For a discussion of vertex roles, see vertexRoles().

Note that annulus index uses faces from tetrahedra index+1 and index+2. The gluing permutation that maps vertices of tetrahedron index+1 to vertices of tetrahedron index+2 will be vertexRoles(index+2) * roleMap * vertexRoles(index+1).inverse().

Parameters

index specifies which annulus on the solid torus boundary to examine; this must be 0, 1 or 2.

Returns: a permutation that describes the gluing of vertex roles, or nullopt if the two triangles of the requested annulus are not glued together.

◆ manifold()

std::unique_ptr< Manifold > regina::TriSolidTorus::manifold ( ) const

overridevirtual

Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented.

If the 3-manifold cannot be recognised then this routine will return null.

The details of which standard triangulations have 3-manifold recognition routines can be found in the notes for the corresponding subclasses of StandardTriangulation. The default implementation of this routine returns null.

It is expected that the number of triangulations whose underlying 3-manifolds can be recognised will grow between releases.

Returns: the underlying 3-manifold.

Reimplemented from regina::StandardTriangulation.

◆ name()

std::string regina::StandardTriangulation::name ( ) const

inherited

Returns the name of this specific triangulation as a human-readable string.

Returns: the name of this triangulation.

◆ operator!=()

bool regina::TriSolidTorus::operator!= ( const TriSolidTorus & other ) const

inline

Determines whether this and the given object represent different specific presentations of a triangular solid torus.

Unlike the parameterised subclasses of StandardTriangulation, this TriSolidTorus subclass represents a fixed structure, and so its comparisons test not for the structure but the precise location of this structure within the enclosing triangulation.

Specifically, two triangular solid tori will compare as equal if and only if each uses the same three numbered tetrahedra, in the same order, and with the same vertex roles. That is, the corresponding permutations returned by vertexRoles() must be equal, and the corresponding tetrahedra returned by tetrahedron() must have equal indices within the triangulation. In particular, it is still meaningful to compare triangular solid tori within different triangulations.

Parameters

other the triangular solid torus to compare with this.

Returns: true if and only if this and the given object represent different specific presentations of a triangular solid torus.

◆ operator=()

TriSolidTorus & regina::TriSolidTorus::operator= ( const TriSolidTorus & )

default

Sets this to be a copy of the given structure.

Returns: a reference to this structure.

◆ operator==()

bool regina::TriSolidTorus::operator== ( const TriSolidTorus & other ) const

inline

Determines whether this and the given object represent the same specific presentation of a triangular solid torus.

Unlike the parameterised subclasses of StandardTriangulation, this TriSolidTorus subclass represents a fixed structure, and so its comparisons test not for the structure but the precise location of this structure within the enclosing triangulation.

Specifically, two triangular solid tori will compare as equal if and only if each uses the same three numbered tetrahedra, in the same order, and with the same vertex roles. That is, the corresponding permutations returned by vertexRoles() must be equal, and the corresponding tetrahedra returned by tetrahedron() must have equal indices within the triangulation. In particular, it is still meaningful to compare triangular solid tori within different triangulations.

Parameters

other the triangular solid torus to compare with this.

Returns: true if and only if this and the given object represent the same specific presentation of a triangular solid torus.

◆ recognise() [1/3]

static std::unique_ptr< StandardTriangulation > regina::StandardTriangulation::recognise ( Component< 3 > * component )

staticinherited

Determines whether the given component represents one of the standard triangulations understood by Regina.

The list of recognised triangulations is expected to grow between releases.

If the standard triangulation returned has boundary triangles then the given component must have the same corresponding boundary triangles, i.e., the component cannot have any further identifications of these boundary triangles with each other.

Note that the triangulation-based routine recognise(const Triangulation<3>&) may recognise more triangulations than this routine, since passing an entire triangulation allows access to more information.

Parameters

component the triangulation component under examination.

Returns: the details of the standard triangulation if the given component is recognised, or null otherwise.

◆ recognise() [2/3]

static std::unique_ptr< StandardTriangulation > regina::StandardTriangulation::recognise ( const Triangulation< 3 > & tri )

staticinherited

Determines whether the given triangulation represents one of the standard triangulations understood by Regina.

The list of recognised triangulations is expected to grow between releases.

If the standard triangulation returned has boundary triangles then the given triangulation must have the same corresponding boundary triangles, i.e., the triangulation cannot have any further identifications of these boundary triangles with each other.

This routine may recognise more triangulations than the component-based recognise(Component<3>*), since passing an entire triangulation allows access to more information.

Parameters

tri	the triangulation under examination.

Returns: the details of the standard triangualation if the given triangulation is recognised, or null otherwise.

◆ recognise() [3/3]

static std::unique_ptr< TriSolidTorus > regina::TriSolidTorus::recognise	(	Tetrahedron< 3 > *	tet,
		Perm< 4 >	useVertexRoles
	)

static

Determines if the given tetrahedron forms part of a three-tetrahedron triangular solid torus with its vertices playing the given roles in the solid torus.

Note that the six boundary triangles of the triangular solid torus need not be boundary triangles within the overall triangulation, i.e., they may be identified with each other or with faces of other tetrahedra.

This function returns by (smart) pointer for consistency with StandardTriangulation::recognise(), which makes use of the polymorphic nature of the StandardTriangulation class hierarchy.

Parameters

tet	the tetrahedron to examine.
useVertexRoles	a permutation describing the role each tetrahedron vertex must play in the solid torus; this must be in the same format as the permutation returned by vertexRoles().

Returns: a structure containing details of the solid torus with the given tetrahedron as tetrahedron 0, or null if the given tetrahedron is not part of a triangular solid torus with the given vertex roles.

◆ str()

std::string regina::Output< StandardTriangulation , 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.

◆ swap()

void regina::TriSolidTorus::swap ( TriSolidTorus & other )

inlinenoexcept

Swaps the contents of this and the given structure.

Parameters

other the structure whose contents should be swapped with this.

◆ tetrahedron()

Tetrahedron< 3 > * regina::TriSolidTorus::tetrahedron ( int index ) const

inline

Returns the requested tetrahedron in this solid torus.

See the general class notes for further details.

Parameters

index specifies which tetrahedron in the solid torus to return; this must be 0, 1 or 2.

Returns: the requested tetrahedron.

◆ texName()

std::string regina::StandardTriangulation::texName ( ) const

inherited

Returns the name of this specific triangulation in TeX format.

No leading or trailing dollar signs will be included.

Warning: The behaviour of this routine has changed as of Regina 4.3; in earlier versions, leading and trailing dollar signs were provided.

Returns: the name of this triangulation in TeX format.

◆ utf8()

std::string regina::Output< StandardTriangulation , 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.

◆ vertexRoles()

Perm< 4 > regina::TriSolidTorus::vertexRoles ( int index ) const

inline

Returns a permutation represeting the role that each vertex of the requested tetrahedron plays in the solid torus.

The permutation returned (call this p) maps 0, 1, 2 and 3 to the four vertices of tetrahedron index so that the edge from p[0] to p[3] is an oriented axis edge, and the path from vertices p[0] to p[1] to p[2] to p[3] follows the three oriented major edges. In particular, the major edge for annulus index will run from vertices p[1] to p[2]. Edges p[0] to p[2] and p[1] to p[3] will both be oriented minor edges.

Note that annulus index+1 uses face p[1] of the requested tetrahedron and annulus index+2 uses face p[2] of the requested tetrahedron. Both annuli use the axis edge p[0] to p[3], and each annulus uses one other major edge and one other minor edge so that (according to homology) the axis edge equals the major edge plus the minor edge.

See the general class notes for further details.

Parameters

index specifies which tetrahedron in the solid torus to examine; this must be 0, 1 or 2.

Returns: a permutation representing the roles of the vertices of the requested tetrahedron.

◆ writeName()

std::ostream & regina::TriSolidTorus::writeName ( std::ostream & out ) const

inlineoverridevirtual

Writes the name of this triangulation as a human-readable string to the given output stream.

Python: Not present. Instead use the variant name() that takes no arguments and returns a string.

Parameters

out	the output stream to which to write.

Returns: a reference to the given output stream.

Implements regina::StandardTriangulation.

◆ writeTeXName()

std::ostream & regina::TriSolidTorus::writeTeXName ( std::ostream & out ) const

inlineoverridevirtual

Writes the name of this triangulation in TeX format to the given output stream.

No leading or trailing dollar signs will be included.

Warning: The behaviour of this routine has changed as of Regina 4.3; in earlier versions, leading and trailing dollar signs were provided.

Python: Not present. Instead use the variant texName() that takes no arguments and returns a string.

Parameters

out	the output stream to which to write.

Returns: a reference to the given output stream.

Implements regina::StandardTriangulation.

◆ writeTextLong()

void regina::StandardTriangulation::writeTextLong ( std::ostream & out ) const

inlinevirtualinherited

Writes a detailed text representation of this object to the given output stream.

This may be reimplemented by subclasses, but the parent StandardTriangulation class offers a reasonable default implementation based on writeTextShort().

Python: Not present. Use detail() instead.

Parameters

out	the output stream to which to write.

Reimplemented in regina::AugTriSolidTorus, regina::BlockedSFS, regina::BlockedSFSLoop, regina::BlockedSFSPair, regina::BlockedSFSTriple, regina::LayeredChainPair, regina::LayeredLensSpace, regina::LayeredLoop, regina::LayeredTorusBundle, regina::PluggedTorusBundle, regina::PlugTriSolidTorus, and regina::TrivialTri.

◆ writeTextShort()

void regina::TriSolidTorus::writeTextShort ( std::ostream & out ) const

overridevirtual

Writes a short text representation of this object to the given output stream.

This may be reimplemented by subclasses, but the parent StandardTriangulation class offers a reasonable default implementation based on writeName().

Python: Not present. Use str() instead.

Parameters

out	the output stream to which to write.

Reimplemented from regina::StandardTriangulation.

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

subcomplex/trisolidtorus.h

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ TriSolidTorus()

Member Function Documentation

◆ areAnnuliLinkedAxis()

◆ areAnnuliLinkedMajor()

◆ detail()

◆ homology()

◆ isAnnulusSelfIdentified()

◆ manifold()

◆ name()

◆ operator!=()

◆ operator=()

◆ operator==()

◆ recognise() [1/3]

◆ recognise() [2/3]

◆ recognise() [3/3]

◆ str()

◆ swap()

◆ tetrahedron()

◆ texName()

◆ utf8()

◆ vertexRoles()

◆ writeName()

◆ writeTeXName()

◆ writeTextLong()

◆ writeTextShort()