Offers routines for constructing a variety of sample 2-dimensional triangulations. More...

#include <triangulation/example2.h>

Inheritance diagram for regina::Example< 2 >:

Static Public Member Functions
static Triangulation< 2 >	orientable (unsigned genus, unsigned punctures)
	Returns a triangulation of the given orientable surface.

static Triangulation< 2 >	nonOrientable (unsigned genus, unsigned punctures)
	Returns a triangulation of the given non-orientable surface.

static Triangulation< 2 >	sphereTetrahedron ()
	Returns the four-triangle 2-sphere formed from the boundary of a tetrahedron.

static Triangulation< 2 >	sphereOctahedron ()
	Returns the eight-triangle 2-sphere formed from the boundary of an octahedron.

static Triangulation< 2 >	disc ()
	Returns a one-triangle disc.

static Triangulation< 2 >	annulus ()
	Returns a two-triangle annulus.

static Triangulation< 2 >	mobius ()
	Returns a one-triangle Mobius band.

static Triangulation< 2 >	torus ()
	Returns a two-triangle torus.

static Triangulation< 2 >	rp2 ()
	Returns a two-triangle projective plane.

static Triangulation< 2 >	kb ()
	Returns a two-triangle Klein bottle.

static Triangulation< dim >	sphere ()
	Closed Triangulations.

static Triangulation< dim >	simplicialSphere ()
	Returns the standard (dim+2)-simplex triangulation of the dim-sphere as the boundary of a (dim+1)-simplex.

static Triangulation< dim >	sphereBundle ()
	Returns a two-simplex triangulation of the product space `S^(dim-1) × S¹`.

static Triangulation< dim >	twistedSphereBundle ()
	Returns a two-simplex triangulation of the twisted product space `S^(dim-1) x~ S¹`.

static Triangulation< dim >	ball ()
	Bounded Triangulations.

static Triangulation< dim >	ballBundle ()
	Returns a triangulation of the product space `B^(dim-1) × S¹`.

static Triangulation< dim >	twistedBallBundle ()
	Returns a triangulation of the twisted product space `B^(dim-1) x~ S¹`.

static Triangulation< dim >	doubleCone (const Triangulation< dim-1 > &base)
	Returns a double cone over the given (dim-1)-dimensional triangulation.

static Triangulation< dim >	singleCone (const Triangulation< dim-1 > &base)
	Returns a single cone over the given (dim-1)-dimensional triangulation.

Detailed Description

Offers routines for constructing a variety of sample 2-dimensional triangulations.

This is a specialisation of the generic Example class template; see the generic Example template documentation for a general overview of how the example triangulation classes work. In Python, you can read this generic documentation by looking at a higher dimension: try help(Example5).

This 2-dimensional specialisation offers significant extra functionality, by providing several more hard-coded constructions.

Member Function Documentation

◆ annulus()

Triangulation< 2 > regina::Example< 2 >::annulus ( )

inlinestatic

Returns a two-triangle annulus.

This is identical to the triangulation returned by the generic routine ballBundle().

Returns: the annulus.

◆ ball()

Triangulation< dim > regina::detail::ExampleBase< dim >::ball ( )

staticinherited

Bounded Triangulations.

Returns a one-simplex triangulation of the dim-ball.

Returns: a one-simplex dim-ball.

◆ ballBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::ballBundle ( )

staticinherited

Returns a triangulation of the product space B^(dim-1) × S¹.

In odd dimensions this will use one simplex, and will therefore be oriented.
In even dimensions this will use two simplices, and will be built as the double cover of the one-simplex B^(dim-1) x~ S¹. The labelling is chosen to highlight this structure, and so even though the space is orientable, the resulting triangulation will not be oriented.

Returns: the product B^(dim-1) × S¹.

◆ disc()

Triangulation< 2 > regina::Example< 2 >::disc ( )

inlinestatic

Returns a one-triangle disc.

This is identical to the triangulation returned by the generic routine ball().

Returns: the disc.

◆ doubleCone()

Triangulation< dim > regina::detail::ExampleFromLowDim< dim, available >::doubleCone ( const Triangulation< dim-1 > & base )

staticinherited

Returns a double cone over the given (dim-1)-dimensional triangulation.

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

Note that, as a special case, if M is either a sphere or a ball, then this routine returns a (dim)-sphere or a (dim)-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 × I we "remove" the vertices at the apex of each cone.

If the given 3-dimensional triangulation is oriented, then the resulting 4-dimensional triangulation will be oriented also.

Warning: If the given (dim-1)-dimensional triangulation has any boundary whatsoever (either real or ideal), then unless it is a (dim-1)-ball, you will obtain an invalid dim-manifold triangulation as a result.

Returns: a double cone over the given triangulation.

◆ kb()

Triangulation< 2 > regina::Example< 2 >::kb ( )

inlinestatic

Returns a two-triangle Klein bottle.

This is identical to the triangulation returned by the generic routine twistedSphereBundle().

Returns: the Klein bottle.

◆ mobius()

Triangulation< 2 > regina::Example< 2 >::mobius ( )

inlinestatic

Returns a one-triangle Mobius band.

This is identical to the triangulation returned by the generic routine twistedBallBundle().

Returns: the Mobius band.

◆ nonOrientable()

static Triangulation< 2 > regina::Example< 2 >::nonOrientable	(	unsigned	genus,
		unsigned	punctures )

static

Returns a triangulation of the given non-orientable surface.

If the number of punctures is 0 or 1, then the resulting triangulation will be minimal (which, with the exception of the projective plane, means there is exactly one vertex).

Parameters

genus	the non-orientable genus of the surface, i.e., the number of crosscaps that it contains; this must be greater than or equal to one.
punctures	the number of punctures in the surface; this must be greater than or equal to zero.

Returns: the requested non-orientable surface.

Author: Alex He, B.B.

◆ orientable()

static Triangulation< 2 > regina::Example< 2 >::orientable	(	unsigned	genus,
		unsigned	punctures )

static

Returns a triangulation of the given orientable surface.

If the number of punctures is 0, then the resulting triangulation will be minimal (which, for positive genus, means there is exactly one vertex).

Parameters

genus	the genus of the surface; this must be greater than or equal to zero.
punctures	the number of punctures in the surface; this must be greater than or equal to zero.

Returns: the requested orientable surface.

◆ rp2()

static Triangulation< 2 > regina::Example< 2 >::rp2 ( )

static

Returns a two-triangle projective plane.

Returns: the projective plane.

◆ simplicialSphere()

Triangulation< dim > regina::detail::ExampleBase< dim >::simplicialSphere ( )

staticinherited

Returns the standard (dim+2)-simplex triangulation of the dim-sphere as the boundary of a (dim+1)-simplex.

Note that the current construction does not give an oriented triangulation (due to the specific choice of labelling); this may change in a future version of Regina.

Returns: the standard simplicial dim-sphere.

◆ singleCone()

Triangulation< dim > regina::detail::ExampleFromLowDim< dim, available >::singleCone ( const Triangulation< dim-1 > & base )

staticinherited

Returns a single cone over the given (dim-1)-dimensional triangulation.

If the given triangulation represents the manifold M, then this returns a triangulation of the product M × 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 (dim-1)-dimensional triangulation base.

If the given 3-dimensional triangulation is oriented, then the resulting 4-dimensional triangulation will be oriented also.

Warning: If the given (dim-1)-dimensional triangulation has any boundary whatsoever (either real or ideal), then unless it is a (dim-1)-ball, you will obtain an invalid dim-manifold triangulation as a result.

Returns: a single cone over the given triangulation.

◆ sphere()

Triangulation< dim > regina::detail::ExampleBase< dim >::sphere ( )

staticinherited

Closed Triangulations.

Returns a two-simplex triangulation of the dim-sphere.

Although the sphere is orientable, this triangulation will not be oriented since the gluings will all be identity permutations.

Returns: a two-simplex dim-sphere.

◆ sphereBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::sphereBundle ( )

staticinherited

Returns a two-simplex triangulation of the product space S^(dim-1) × S¹.

Note that the current construction does not give an oriented triangulation (due to the specific choice of labelling); this may change in a future version of Regina.

Returns: the product S^(dim-1) × S¹.

◆ sphereOctahedron()

static Triangulation< 2 > regina::Example< 2 >::sphereOctahedron ( )

static

Returns the eight-triangle 2-sphere formed from the boundary of an octahedron.

Returns: the octahedral sphere.

◆ sphereTetrahedron()

Triangulation< 2 > regina::Example< 2 >::sphereTetrahedron ( )

inlinestatic

Returns the four-triangle 2-sphere formed from the boundary of a tetrahedron.

This is identical to the triangulation returned by the generic routine simplicialSphere().

Returns: the tetrahedral sphere.

◆ torus()

Triangulation< 2 > regina::Example< 2 >::torus ( )

inlinestatic

Returns a two-triangle torus.

This is identical to the triangulation returned by the generic routine sphereBundle().

Returns: the torus.

◆ twistedBallBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::twistedBallBundle ( )

staticinherited

Returns a triangulation of the twisted product space B^(dim-1) x~ S¹.

This will use one simplex in even dimensions, or two simplices in odd dimensions.

Returns: the twisted product B^(dim-1) x~ S¹.

◆ twistedSphereBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::twistedSphereBundle ( )

staticinherited

Returns a two-simplex triangulation of the twisted product space S^(dim-1) x~ S¹.

Returns: the twisted product S^(dim-1) x~ S¹.

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

triangulation/example2.h

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ annulus()

◆ ball()

◆ ballBundle()

◆ disc()

◆ doubleCone()

◆ kb()

◆ mobius()

◆ nonOrientable()

◆ orientable()

◆ rp2()

◆ simplicialSphere()

◆ singleCone()

◆ sphere()

◆ sphereBundle()

◆ sphereOctahedron()

◆ sphereTetrahedron()

◆ torus()

◆ twistedBallBundle()

◆ twistedSphereBundle()