Regina Calculation Engine
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regina::Dim4ExampleTriangulation Class Reference

This class offers routines for constructing a variety of sample 4-manifold triangulations. More...

#include <dim4/dim4exampletriangulation.h>

Static Public Member Functions

Closed Triangulations
static Dim4TriangulationfourSphere ()
 Returns a two-pentachoron triangulation of the 4-sphere. More...
 
static Dim4TriangulationsimplicialFourSphere ()
 Returns the standard six-pentachoron triangulation of the 4-sphere as the boundary of a 5-simplex. More...
 
static Dim4Triangulationrp4 ()
 Returns a four-pentachoron triangulation of real projective 4-space. More...
 
static Dim4Triangulations3xs1 ()
 Returns a two-pentachoron triangulation of the product space S^3 x S^1. More...
 
static Dim4Triangulations3xs1Twisted ()
 Returns a two-pentachoron triangulation of the twisted product space S^3 x~ S^1. More...
 
Ideal Triangulations

(end: Closed Triangulations)

static Dim4TriangulationcappellShaneson ()
 Returns a two-pentachoron triangulation of a Cappell-Shaneson 2-knot complement in the 4-sphere. More...
 
Constructions from 3-Manifold Triangulations

(end: Ideal Triangulations)

static Dim4TriangulationdoubleCone (const NTriangulation &base)
 Returns a double cone over the given 3-manifold triangulation. More...
 
static Dim4TriangulationsingleCone (const NTriangulation &base)
 Returns a single cone over the given 3-manifold triangulation. More...
 
static Dim4TriangulationiBundle (const NTriangulation &base)
 Returns a triangulation of the product M x I, where M is the given 3-manifold triangulation. More...
 
static Dim4Triangulations1Bundle (const NTriangulation &base)
 Returns a triangulation of the product M x S1, where M is the given 3-manifold triangulation. More...
 
static Dim4TriangulationbundleWithMonodromy (const NTriangulation &base, const NIsomorphism &monodromy)
 Returns a bundle formed from a given 3-manifold and a given monodromy. More...
 

Detailed Description

This class offers routines for constructing a variety of sample 4-manifold triangulations.

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

At the present time only a handful of triangulations are available (in contrast to the 3-dimensional NExampleTriangulation, which offers many interesting examples). The authors hope to expand this list of 4-dimensional examples in future releases of Regina.

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

§ bundleWithMonodromy()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::bundleWithMonodromy ( const NTriangulation base,
const NIsomorphism monodromy 
)
static

Returns a bundle formed from a given 3-manifold and a given monodromy.

Specifically, let M be the given 3-manifold triangulation. This routine builds the bundle M x I, and then identifies the two copies of M on the boundary according to the given homeomorphism from M to itself. The homeomorphism must be expressed as a combinatorial automorphism, which means that for a non-trivial monodromy you may need to do some work to find a sufficiently symmetric 3-manifold triangulation to begin with.

The resulting manifold will contain 82 pentachora for each original tetrahedron of M, and will contain many internal vertices. It is highly recommended that you call Dim4Triangulation::intelligentSimplify() afterwards if you do not need to preserve the combinatorial structure.

Precondition
The given monodromy must be an isomorphism from M to itself; that is, a combinatorial automorphism.
Warning
If the given 3-manifold triangulation has ideal boundary, then you will obtain an invalid 4-manifold triangulation as a result.
Parameters
basethe 3-manifold triangulation M, as described above.
monodromythe homeomorphism from M to itself, as described above.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ cappellShaneson()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::cappellShaneson ( )
static

Returns a two-pentachoron triangulation of a Cappell-Shaneson 2-knot complement in the 4-sphere.

This triangulation is described and analysed in "Triangulating a Cappell-Shaneson knot complement", Budney, Burton and Hillman, Mathematical Research Letters 19 (2012), no. 5, 1117-1126.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ doubleCone()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::doubleCone ( const NTriangulation base)
static

Returns a double cone over the given 3-manifold triangulation.

If the given 3-manifold is M, this returns an ideal triangulation of the product M x I (with two ideal boundary components). A copy of the original 3-manifold triangulation base can be found at the centre of this construction, formed from the tetrahedra that sit between the two ideal vertices.

Note that, as a special case, if M is either a 3-sphere or a 3-ball, then this routine returns a 4-sphere or a 4-ball (since "ideal spheres" and "ideal balls" just become regular internal and boundary vertices respectively).

This construction is essentially the suspension of the triangulation base. We do not call it this however, since from a topological point of view, to form the ideal triangulation of M x I we "remove" the vertices at the apex of each cone.

Warning
If the given 3-manifold triangulation has any boundary whatsoever (either real or ideal), then unless this 3-manifold triangulation is a 3-ball you will obtain an invalid 4-manifold triangulation as a result.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ fourSphere()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::fourSphere ( )
static

Returns a two-pentachoron triangulation of the 4-sphere.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ iBundle()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::iBundle ( const NTriangulation base)
static

Returns a triangulation of the product M x I, where M is the given 3-manifold triangulation.

The boundary of this product will consist of two copies of M, both combinatorially isomorphic to the original triangulation. If n is the number of tetrahedra in M, then the first copy of M on the boundary is obtained by mapping vertices 0,1,2,3 of tetrahedron i of M to vertices 0,1,2,3 of pentachoron i, and the second copy is obtained by mapping vertices 0,1,2,3 of tetrahedron i of M to vertices 0,1,2,3 of pentachoron n+i.

The product itself will contain 82 pentachora for each original tetrahedron of M, and will contain many internal vertices. It is highly recommended that you call Dim4Triangulation::intelligentSimplify() afterwards if you do not need to preserve the combinatorial structure.

Warning
If the given 3-manifold triangulation has ideal boundary, then you will obtain an invalid 4-manifold triangulation as a result.
Parameters
basethe 3-manifold triangulation M, as described above.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ rp4()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::rp4 ( )
static

Returns a four-pentachoron triangulation of real projective 4-space.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.
Author
Ryan Budney

§ s1Bundle()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::s1Bundle ( const NTriangulation base)
static

Returns a triangulation of the product M x S1, where M is the given 3-manifold triangulation.

This simply calls iBundle() and then glues together the two copies of M on the boundary.

The product will contain 82 pentachora for each original tetrahedron of M, and will contain many internal vertices. It is highly recommended that you call Dim4Triangulation::intelligentSimplify() afterwards if you do not need to preserve the combinatorial structure.

Warning
If the given 3-manifold triangulation has ideal boundary, then you will obtain an invalid 4-manifold triangulation as a result.
Parameters
basethe 3-manifold triangulation M, as described above.
Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ s3xs1()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::s3xs1 ( )
static

Returns a two-pentachoron triangulation of the product space S^3 x S^1.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ s3xs1Twisted()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::s3xs1Twisted ( )
static

Returns a two-pentachoron triangulation of the twisted product space S^3 x~ S^1.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ simplicialFourSphere()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::simplicialFourSphere ( )
static

Returns the standard six-pentachoron triangulation of the 4-sphere as the boundary of a 5-simplex.

Returns
a newly constructed triangulation, which must be destroyed by the caller of this routine.

§ singleCone()

static Dim4Triangulation* regina::Dim4ExampleTriangulation::singleCone ( const NTriangulation base)
static

Returns a single cone over the given 3-manifold triangulation.

If the given 3-manifold is M, this returns a triangulation of the product M x I that has one real boundary component and one ideal boundary component. The triangulation of the real boundary component will be identical to the original 3-manifold triangulation base.

Warning
If the given 3-manifold triangulation has any boundary whatsoever (either real or ideal), then unless this 3-manifold triangulation is a 3-ball you will obtain an invalid 4-manifold triangulation as a result.
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).