Regina Calculation Engine
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regina::NExampleTriangulation Class Reference

This class offers routines for constructing sample 3-manifold triangulations of various types. More...

#include <triangulation/nexampletriangulation.h>

Static Public Member Functions

Closed Triangulations
static NTriangulationthreeSphere ()
 Returns a one-tetrahedron triangulation of the 3-sphere. More...
 
static NTriangulationbingsHouse ()
 Returns the two-tetrahedron triangulation of the 3-sphere that is dual to Bing's house with two rooms. More...
 
static NTriangulations2xs1 ()
 Returns a two-tetrahedron triangulation of the product space S^2 x S^1. More...
 
static NTriangulationrp2xs1 ()
 Returns a three-tetrahedron triangulation of the non-orientable product space RP^2 x S^1. More...
 
static NTriangulationrp3rp3 ()
 Returns a triangulation of the connected sum RP^3 # RP^3. More...
 
static REGINA_DEPRECATED
NTriangulation
lens8_3 ()
 Deprecated routine that returns the minimal triangulation of the lens space L(8,3). More...
 
static NTriangulationlens (size_t p, size_t q)
 Returns a triangulation of the lens space L(p,q). More...
 
static NTriangulationpoincareHomologySphere ()
 Returns the five-tetrahedron triangulation of the Poincare homology sphere. More...
 
static NTriangulationweeks ()
 Returns a nine-tetrahedron minimal triangulation of the Weeks manifold. More...
 
static NTriangulationweberSeifert ()
 Returns a one-vertex triangulation of the Weber-Seifert dodecahedral space. More...
 
static REGINA_DEPRECATED
NTriangulation
seifertWeber ()
 Returns a one-vertex triangulation of the Weber-Seifert dodecahedral space. More...
 
static NTriangulationsmallClosedOrblHyperbolic ()
 Returns the nine-tetrahedron closed orientable hyperbolic 3-manifold with volume 0.94270736. More...
 
static NTriangulationsmallClosedNonOrblHyperbolic ()
 Returns the eleven-tetrahedron closed non-orientable hyperbolic 3-manifold with volume 2.02988321. More...
 
static NTriangulationsphere600 ()
 Returns the boundary 3-sphere of the regular 600-cell. More...
 
Finite Bounded Triangulations
static REGINA_DEPRECATED
NTriangulation
lst3_4_7 ()
 Deprecated routine that returns the three-tetrahedron layered solid torus LST(3,4,7). More...
 
static NTriangulationlst (size_t a, size_t b)
 Returns the layered solid torus LST(a,b,c). More...
 
static NTriangulationsolidKleinBottle ()
 Returns a triangulation of the solid Klein bottle. More...
 
Ideal Triangulations
static NTriangulationfigureEight ()
 Returns a two-tetrahedron ideal triangulation of the figure eight knot complement. More...
 
static REGINA_DEPRECATED
NTriangulation
figureEightKnotComplement ()
 Deprecated function that teturns a two-tetrahedron ideal triangulation of the figure eight knot complement. More...
 
static NTriangulationtrefoil ()
 Returns a two-tetrahedron ideal triangulation of the trefoil knot complement. More...
 
static REGINA_DEPRECATED
NTriangulation
trefoilKnotComplement ()
 Deprecated function that returns a two-tetrahedron ideal triangulation of the trefoil knot complement. More...
 
static NTriangulationwhiteheadLink ()
 Returns a four-tetrahedron ideal triangulation of the Whitehead link complement. More...
 
static REGINA_DEPRECATED
NTriangulation
whiteheadLinkComplement ()
 Deprecated function that returns a four-tetrahedron ideal triangulation of the Whitehead link complement. More...
 
static NTriangulationgieseking ()
 Returns the one-tetrahedron ideal triangulation of the non-orientable Gieseking manifold. More...
 
static NTriangulationcuspedGenusTwoTorus ()
 Returns a triangulation of a solid genus two torus with a cusped boundary. More...
 

Detailed Description

This class offers routines for constructing sample 3-manifold triangulations of various types.

These triangulations may be useful for testing new code, or for simply getting a feel for how Regina works.

The sample triangulations offered here may prove especially useful in Regina's scripting interface, where working with pre-existing files is more complicated than in the GUI.

All of the methods in this class will assign an appropriate packet label to the triangulation that they return.

Note that each of these routines constructs a new triangulation from scratch. It is up to the caller of each routine to destroy the triangulation that is returned.

Member Function Documentation

static NTriangulation* regina::NExampleTriangulation::bingsHouse ( )
static

Returns the two-tetrahedron triangulation of the 3-sphere that is dual to Bing's house with two rooms.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::cuspedGenusTwoTorus ( )
static

Returns a triangulation of a solid genus two torus with a cusped boundary.

This triangulation has one internal finite vertex and one genus two ideal vertex.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::figureEight ( )
static

Returns a two-tetrahedron ideal triangulation of the figure eight knot complement.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
NTriangulation * regina::NExampleTriangulation::figureEightKnotComplement ( )
inlinestatic

Deprecated function that teturns a two-tetrahedron ideal triangulation of the figure eight knot complement.

Deprecated:
This has been renamed to figureEight().
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::gieseking ( )
static

Returns the one-tetrahedron ideal triangulation of the non-orientable Gieseking manifold.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::lens ( size_t  p,
size_t  q 
)
static

Returns a triangulation of the lens space L(p,q).

The triangulation uses a layered lens space, which is conjectured (but not proven in all cases) to be the triangulation requiring the fewest tetrahedra.

Precondition
p > q >= 0 unless (p,q) = (0,1).
gcd(p, q) = 1.
Parameters
pa parameter of the desired lens space.
qa parameter of the desired lens space.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static REGINA_DEPRECATED NTriangulation* regina::NExampleTriangulation::lens8_3 ( )
static

Deprecated routine that returns the minimal triangulation of the lens space L(8,3).

Deprecated:
This is identical to calling NExampleTriangulation::lens(8, 3). This hard-coded lens8_3() routine will be removed in a future release of Regina.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::lst ( size_t  a,
size_t  b 
)
static

Returns the layered solid torus LST(a,b,c).

This is a parameterised triangulation of the solid torus. It has two boundary triangles and three boundary edges, and the meridional disc of the solid torus cuts these boundary edges a, b and c times respectively.

Only the parameters a and b are passed as arguments to this routine. The third parameter c will be deduced automatically as c = (a + b).

Precondition
gcd(a, b) = 1.
Parameters
athe first parameter of the layered solid torus.
bthe second parameter of the layered solid torus.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static REGINA_DEPRECATED NTriangulation* regina::NExampleTriangulation::lst3_4_7 ( )
static

Deprecated routine that returns the three-tetrahedron layered solid torus LST(3,4,7).

Deprecated:
This is identical to calling NExampleTriangulation::lst(3, 4). This hard-coded lst3_4_7() routine will be removed in a future release of Regina.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::poincareHomologySphere ( )
static

Returns the five-tetrahedron triangulation of the Poincare homology sphere.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::rp2xs1 ( )
static

Returns a three-tetrahedron triangulation of the non-orientable product space RP^2 x S^1.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::rp3rp3 ( )
static

Returns a triangulation of the connected sum RP^3 # RP^3.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::s2xs1 ( )
static

Returns a two-tetrahedron triangulation of the product space S^2 x S^1.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static REGINA_DEPRECATED NTriangulation* regina::NExampleTriangulation::seifertWeber ( )
static

Returns a one-vertex triangulation of the Weber-Seifert dodecahedral space.

Deprecated:
This routine is now called weberSeifert(), for consistency with Weber and Seifert's original paper. The old name seifertWeber() has been kept for backward compatibility, but will be removed in a future version of Regina.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::smallClosedNonOrblHyperbolic ( )
static

Returns the eleven-tetrahedron closed non-orientable hyperbolic 3-manifold with volume 2.02988321.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::smallClosedOrblHyperbolic ( )
static

Returns the nine-tetrahedron closed orientable hyperbolic 3-manifold with volume 0.94270736.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::solidKleinBottle ( )
static

Returns a triangulation of the solid Klein bottle.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::sphere600 ( )
static

Returns the boundary 3-sphere of the regular 600-cell.

This is a triangulation of the 3-sphere that is a simplicial complex, and in which every edge has degree five.

The triangulation was extracted from the Benedetti-Lutz library of triangulations. See: http://page.math.tu-berlin.de/~lutz/stellar/library_of_triangulations.html

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::threeSphere ( )
static

Returns a one-tetrahedron triangulation of the 3-sphere.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::trefoil ( )
static

Returns a two-tetrahedron ideal triangulation of the trefoil knot complement.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
NTriangulation * regina::NExampleTriangulation::trefoilKnotComplement ( )
inlinestatic

Deprecated function that returns a two-tetrahedron ideal triangulation of the trefoil knot complement.

Deprecated:
This has been renamed to trefoil().
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::weberSeifert ( )
static

Returns a one-vertex triangulation of the Weber-Seifert dodecahedral space.

This 3-manifold is described in "Die beiden Dodekaederraume", C. Weber and H. Seifert, Math. Z. 37 (1933), no. 1, 237-253. The triangulation returned by this routine (with 23 tetrahedra) is given in "The Weber-Seifert dodecahedral space is non-Haken", Benjamin A. Burton, J. Hyam Rubinstein and Stephan Tillmann, Trans. Amer. Math. Soc. 364:2 (2012), pp. 911-932.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::weeks ( )
static

Returns a nine-tetrahedron minimal triangulation of the Weeks manifold.

The Weeks manifold is the smallest-volume closed hyperbolic 3-manifold, with a volume of roughly 0.9427. Note that there are nine minimal triangulations of the Weeks manifold (of course this routine returns just one).

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
static NTriangulation* regina::NExampleTriangulation::whiteheadLink ( )
static

Returns a four-tetrahedron ideal triangulation of the Whitehead link complement.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
NTriangulation * regina::NExampleTriangulation::whiteheadLinkComplement ( )
inlinestatic

Deprecated function that returns a four-tetrahedron ideal triangulation of the Whitehead link complement.

Deprecated:
This has been renamed to whiteheadLink().
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

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

Copyright © 1999-2016, 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).