Regina Calculation Engine
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regina::SpiralSolidTorus Class Reference

Represents a spiralled solid torus in a triangulation. More...

#include <subcomplex/spiralsolidtorus.h>

Inheritance diagram for regina::SpiralSolidTorus:
regina::StandardTriangulation regina::Output< StandardTriangulation >

Public Member Functions

virtual ~SpiralSolidTorus ()
 Destroys this spiralled solid torus. More...
 
SpiralSolidTorusclone () const
 Returns a newly created clone of this structure. More...
 
size_t size () const
 Returns the number of tetrahedra in this spiralled solid torus. More...
 
Tetrahedron< 3 > * tetrahedron (size_t index) const
 Returns the requested tetrahedron in this spiralled solid torus. More...
 
Perm< 4 > vertexRoles (size_t index) const
 Returns a permutation represeting the role that each vertex of the requested tetrahedron plays in the solid torus. More...
 
void reverse ()
 Reverses this spiralled solid torus. More...
 
void cycle (size_t k)
 Cycles this spiralled solid torus by the given number of tetrahedra. More...
 
bool makeCanonical (const Triangulation< 3 > *tri)
 Converts this spiralled solid torus into its canonical representation. More...
 
bool isCanonical (const Triangulation< 3 > *tri) const
 Determines whether this spiralled solid torus is in canonical form. More...
 
Manifoldmanifold () const override
 Returns the 3-manifold represented by this triangulation, if such a recognition routine has been implemented. More...
 
AbelianGrouphomology () 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 writeTextLong (std::ostream &out) const override
 
Writes a detailed 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...
 
AbelianGrouphomologyH1 () const
 Returns the expected first homology group of this triangulation, if such a routine has been implemented. More...
 
virtual void writeTextShort (std::ostream &out) const
 
Writes a short 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 SpiralSolidTorusformsSpiralSolidTorus (Tetrahedron< 3 > *tet, Perm< 4 > useVertexRoles)
 Determines if the given tetrahedron forms part of a spiralled solid torus with its vertices playing the given roles in the solid torus. More...
 
static StandardTriangulationisStandardTriangulation (Component< 3 > *component)
 Determines whether the given component represents one of the standard triangulations understood by Regina. More...
 
static StandardTriangulationisStandardTriangulation (Triangulation< 3 > *tri)
 Determines whether the given triangulation represents one of the standard triangulations understood by Regina. More...
 

Detailed Description

Represents a spiralled solid torus in a triangulation.

A spiralled solid torus is created by placing tetrahedra one upon another in a spiralling fashion to form a giant loop.

For each tetrahedron, label the vertices A, B, C and D. Draw the tetrahedron so that the vertices form an upward spiral in the order A-B-C-D, with D directly above A. Face BCD is on the top, face ABC is on the bottom and faces ABD and ACD are both vertical.

When joining two tetrahedra, face BCD of the lower tetrahedron will be joined to face ABC of the upper tetrahedron. In this way the tetrahedra are placed one upon another to form a giant loop (which is closed up by placing the bottommost tetrahedron above the topmost tetrahedron in a similar fashion), forming a solid torus overall.

In each tetrahedron, directed edges AB, BC and CD are major edges, directed edges AC and BD are minor edges and directed edge AD is an axis edge.

The major edges all combined form a single longitude of the solid torus. Using this directed longitude, using the directed meridinal curve ACBA and assuming the spiralled solid torus contains n tetrahedra, the minor edges all combined form a (2, n) curve and the axis edges all combined form a (3, n) curve on the torus boundary.

Note that all tetrahedra in the spiralled solid torus must be distinct and there must be at least one tetrahedron.

Note also that class TriSolidTorus represents a spiralled solid torus with precisely three tetrahedra. A spiralled solid torus with only one tetrahedron is in fact a (1,2,3) layered solid torus.

All optional StandardTriangulation routines are implemented for this class.

Constructor & Destructor Documentation

◆ ~SpiralSolidTorus()

regina::SpiralSolidTorus::~SpiralSolidTorus ( )
inlinevirtual

Destroys this spiralled solid torus.

Member Function Documentation

◆ clone()

SpiralSolidTorus* regina::SpiralSolidTorus::clone ( ) const

Returns a newly created clone of this structure.

Returns
a newly created clone.

◆ cycle()

void regina::SpiralSolidTorus::cycle ( size_t  k)

Cycles this spiralled solid torus by the given number of tetrahedra.

Tetrahedra k, k+1, k+2 and so on will become tetrahedra 0, 1, 2 and so on respectively. Note that this operation will not change the vertex roles.

The underlying triangulation is not changed; all that changes is how this spiralled solid torus is represented.

Parameters
kthe number of tetrahedra through which we should cycle.

◆ 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.

◆ formsSpiralSolidTorus()

static SpiralSolidTorus* regina::SpiralSolidTorus::formsSpiralSolidTorus ( Tetrahedron< 3 > *  tet,
Perm< 4 >  useVertexRoles 
)
static

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

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

Parameters
tetthe tetrahedron to examine.
useVertexRolesa 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 newly created 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 spiralled solid torus with the given vertex roles.

◆ homology()

AbelianGroup* regina::SpiralSolidTorus::homology ( ) const
overridevirtual

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

If the calculation of homology has not yet been implemented for this triangulation then this routine will return 0.

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.

The details of which standard triangulations have homology calculation routines can be found in the notes for the corresponding subclasses of StandardTriangulation. The default implementation of this routine returns 0.

The homology group will be newly allocated and must be destroyed by the caller of this routine.

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.

This routine can also be accessed via the alias homologyH1() (a name that is more specific, but a little longer to type).

Returns
the first homology group of this triangulation, or 0 if the appropriate calculation routine has not yet been implemented.

Reimplemented from regina::StandardTriangulation.

◆ homologyH1()

AbelianGroup * regina::StandardTriangulation::homologyH1 ( ) const
inlineinherited

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

If the calculation of homology has not yet been implemented for this triangulation then this routine will return 0.

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.

The details of which standard triangulations have homology calculation routines can be found in the notes for the corresponding subclasses of StandardTriangulation. The default implementation of this routine returns 0.

The homology group will be newly allocated and must be destroyed by the caller of this routine.

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.

This routine can also be accessed via the alias homology() (a name that is less specific, but a little easier to type).

Returns
the first homology group of this triangulation, or 0 if the appropriate calculation routine has not yet been implemented.

◆ isCanonical()

bool regina::SpiralSolidTorus::isCanonical ( const Triangulation< 3 > *  tri) const

Determines whether this spiralled solid torus is in canonical form.

Canonical form is described in detail in the description for makeCanonical().

Parameters
trithe triangulation in which this solid torus lives.
Returns
true if and only if this spiralled solid torus is in canonical form.

◆ isStandardTriangulation() [1/2]

static StandardTriangulation* regina::StandardTriangulation::isStandardTriangulation ( 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 isStandardTriangulation(Triangulation<3>*) may recognise more triangulations than this routine, since passing an entire triangulation allows access to more information.

Parameters
componentthe triangulation component under examination.
Returns
the details of the standard triangulation if the given component is recognised, or 0 otherwise.

◆ isStandardTriangulation() [2/2]

static StandardTriangulation* regina::StandardTriangulation::isStandardTriangulation ( 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 isStandardTriangulation(Component<3>*), since passing an entire triangulation allows access to more information.

Parameters
trithe triangulation under examination.
Returns
the details of the standard triangualation if the given triangulation is recognised, or 0 otherwise.

◆ makeCanonical()

bool regina::SpiralSolidTorus::makeCanonical ( const Triangulation< 3 > *  tri)

Converts this spiralled solid torus into its canonical representation.

The canonical representation of a spiralled solid torus is unique in a given triangulation.

Tetrahedron 0 in the spiralled solid torus will be the tetrahedron with the lowest index in the triangulation, and under permutation vertexRoles(0) the image of 0 will be less than the image of 3.

Parameters
trithe triangulation in which this solid torus lives.
Returns
true if and only if the representation of this spiralled solid torus was actually changed.

◆ manifold()

Manifold* regina::SpiralSolidTorus::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 0.

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 0.

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

The 3-manifold will be newly allocated and must be destroyed by the caller of this routine.

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.

◆ reverse()

void regina::SpiralSolidTorus::reverse ( )

Reverses this spiralled solid torus.

Tetrahedra 0, 1, 2, ..., size()-1 will become tetrahedra size()-1, ..., 2, 1, 0 respectively. Note that this operation will change the vertex roles as well.

The underlying triangulation is not changed; all that changes is how this spiralled solid torus is represented.

◆ size()

size_t regina::SpiralSolidTorus::size ( ) const
inline

Returns the number of tetrahedra in this spiralled solid torus.

Returns
the number of tetrahedra.

◆ 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 fit on a single line, and should not end with a newline. Where possible, it should use plain ASCII characters.

Python
In addition to str(), this is also used as the Python "stringification" function __str__().
Returns
a short text representation of this object.

◆ tetrahedron()

Tetrahedron< 3 > * regina::SpiralSolidTorus::tetrahedron ( size_t  index) const
inline

Returns the requested tetrahedron in this spiralled solid torus.

Tetrahedra are numbered from 0 to size()-1 inclusive, with tetrahedron i+1 being placed above tetrahedron i.

Parameters
indexspecifies which tetrahedron to return; this must be between 0 and size()-1 inclusive.
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 fit on a single line, and should not end with a newline. In addition, it may use unicode characters to make the output more pleasant to read. This string will be encoded in UTF-8.

Returns
a short text representation of this object.

◆ vertexRoles()

Perm< 4 > regina::SpiralSolidTorus::vertexRoles ( size_t  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 vertices p[0], p[1], p[2] and p[3] correspond to vertices A, B, C and D respectively as described in the general class notes.

In particular, the directed edge from vertex p[0] to p[3] is an axis edge, directed edges p[0] to p[2] and p[1] to p[3] are minor edges and the directed path from vertices p[0] to p[1] to p[2] to p[3] follows the three major edges.

See the general class notes for further details.

Parameters
indexspecifies which tetrahedron in the solid torus to examine; this must be between 0 and size()-1 inclusive.
Returns
a permutation representing the roles of the vertices of the requested tetrahedron.

◆ writeName()

std::ostream & regina::SpiralSolidTorus::writeName ( std::ostream &  out) const
inlineoverridevirtual


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

Python
The parameter out does not exist; standard output will be used.
Parameters
outthe output stream to which to write.
Returns
a reference to the given output stream.

Implements regina::StandardTriangulation.

◆ writeTeXName()

std::ostream & regina::SpiralSolidTorus::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
The parameter out does not exist; standard output will be used.
Parameters
outthe output stream to which to write.
Returns
a reference to the given output stream.

Implements regina::StandardTriangulation.

◆ writeTextLong()

void regina::SpiralSolidTorus::writeTextLong ( std::ostream &  out) const
inlineoverridevirtual


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 writeName().

Python
Not present.
Parameters
outthe output stream to which to write.

Reimplemented from regina::StandardTriangulation.

◆ writeTextShort()

void regina::StandardTriangulation::writeTextShort ( std::ostream &  out) const
inlinevirtualinherited


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.
Parameters
outthe output stream to which to write.

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

Copyright © 1999-2021, The Regina development team
This software is released under the GNU General Public License, with some additional permissions; see the source code for details.
For further information, or to submit a bug or other problem, please contact Ben Burton (bab@maths.uq.edu.au).