Regina 7.0 Calculation Engine
Static Public Member Functions | List of all members
regina::Example< 3 > Class Reference

Offers routines for constructing a variety of sample 3-dimensional triangulations. More...

#include <triangulation/example3.h>

Inheritance diagram for regina::Example< 3 >:
regina::detail::ExampleBase< 3 > regina::detail::ExampleFromLowDim< dim, dim !=2 >

Static Public Member Functions

static Triangulation< dim > sphere ()
 Closed Triangulations. More...
 
static Triangulation< dim > simplicialSphere ()
 Returns the standard (dim+2)-simplex triangulation of the dim-sphere as the boundary of a (dim+1)-simplex. More...
 
static Triangulation< dim > sphereBundle ()
 Returns a two-simplex triangulation of the product space S^(dim-1) x S^1. More...
 
static Triangulation< dim > twistedSphereBundle ()
 Returns a two-simplex triangulation of the twisted product space S^(dim-1) x~ S^1. More...
 
static Triangulation< dim > ball ()
 Bounded Triangulations. More...
 
static Triangulation< dim > ballBundle ()
 Returns a triangulation of the product space B^(dim-1) x S^1. More...
 
static Triangulation< dim > twistedBallBundle ()
 Returns a triangulation of the twisted product space B^(dim-1) x~ S^1. More...
 
static Triangulation< dim > doubleCone (const Triangulation< dim-1 > &base)
 Returns a double cone over the given (dim-1)-dimensional triangulation. More...
 
static Triangulation< dim > singleCone (const Triangulation< dim-1 > &base)
 Returns a single cone over the given (dim-1)-dimensional triangulation. More...
 
Closed Triangulations
static Triangulation< 3 > threeSphere ()
 Returns a one-tetrahedron triangulation of the 3-sphere. More...
 
static Triangulation< 3 > bingsHouse ()
 Returns the two-tetrahedron triangulation of the 3-sphere that is dual to Bing's house with two rooms. More...
 
static Triangulation< 3 > s2xs1 ()
 Returns a two-tetrahedron triangulation of the product space S^2 x S^1. More...
 
static Triangulation< 3 > rp2xs1 ()
 Returns a three-tetrahedron triangulation of the non-orientable product space RP^2 x S^1. More...
 
static Triangulation< 3 > rp3rp3 ()
 Returns a triangulation of the connected sum RP^3 # RP^3. More...
 
static Triangulation< 3 > lens (size_t p, size_t q)
 Returns a triangulation of the lens space L(p,q). More...
 
static Triangulation< 3 > layeredLoop (size_t length, bool twisted)
 Returns a layered loop of the given length. More...
 
static Triangulation< 3 > poincare ()
 Returns the five-tetrahedron triangulation of the Poincare homology sphere. More...
 
static Triangulation< 3 > poincareHomologySphere ()
 Deprecated routine that returns the five-tetrahedron triangulation of the Poincare homology sphere. More...
 
static Triangulation< 3 > augTriSolidTorus (long a1, long b1, long a2, long b2, long a3, long b3)
 Returns an augmented triangular solid torus with the given parameters. More...
 
static Triangulation< 3 > sfsOverSphere (long a1=1, long b1=0, long a2=1, long b2=0, long a3=1, long b3=0)
 Returns a triangulation of the given orientable Seifert fibred space over the sphere with at most three exceptional fibres. More...
 
static Triangulation< 3 > weeks ()
 Returns a nine-tetrahedron minimal triangulation of the Weeks manifold. More...
 
static Triangulation< 3 > weberSeifert ()
 Returns a one-vertex triangulation of the Weber-Seifert dodecahedral space. More...
 
static Triangulation< 3 > smallClosedOrblHyperbolic ()
 Returns the nine-tetrahedron closed orientable hyperbolic 3-manifold with volume 0.94270736. More...
 
static Triangulation< 3 > smallClosedNonOrblHyperbolic ()
 Returns the eleven-tetrahedron closed non-orientable hyperbolic 3-manifold with volume 2.02988321. More...
 
static Triangulation< 3 > sphere600 ()
 Returns the boundary 3-sphere of the regular 600-cell. More...
 
Finite Bounded Triangulations
static Triangulation< 3 > lst (size_t a, size_t b)
 Returns the layered solid torus LST(a,b,c). More...
 
static Triangulation< 3 > solidKleinBottle ()
 Returns a triangulation of the solid Klein bottle. More...
 
Ideal Triangulations
static Triangulation< 3 > figureEight ()
 Returns a two-tetrahedron ideal triangulation of the figure eight knot complement. More...
 
static Triangulation< 3 > trefoil ()
 Returns a two-tetrahedron ideal triangulation of the trefoil knot complement. More...
 
static Triangulation< 3 > whiteheadLink ()
 Returns a four-tetrahedron ideal triangulation of the Whitehead link complement. More...
 
static Triangulation< 3 > gieseking ()
 Returns the one-tetrahedron ideal triangulation of the non-orientable Gieseking manifold. More...
 
static Triangulation< 3 > cuspedGenusTwoTorus ()
 Returns a triangulation of a solid genus two torus with a cusped boundary. More...
 

Detailed Description

Offers routines for constructing a variety of sample 3-dimensional triangulations.

This is a specialisation of the generic Example class template; see the Example template documentation for a general overview of how the example triangulation classes work.

This 3-dimensional specialisation offers significant extra functionality, by providing several more hard-coded and parameterised constructions.

Member Function Documentation

◆ augTriSolidTorus()

static Triangulation< 3 > regina::Example< 3 >::augTriSolidTorus ( long  a1,
long  b1,
long  a2,
long  b2,
long  a3,
long  b3 
)
static

Returns an augmented triangular solid torus with the given parameters.

Almost all augmented triangular solid tori represent Seifert fibred spaces with three or fewer exceptional fibres. Augmented triangular solid tori are described in more detail in the AugTriSolidTorus class notes.

The resulting Seifert fibred space will be SFS((a1, b1), (a2, b2), (a3, b3), (1, 1)), where the parameters a1, ..., b3 are passed as arguments to this routine. The three layered solid tori that are attached to the central triangular solid torus will be LST(|a1|, |b1|, |-a1-b1|), ..., LST(|a3|, |b3|, |-a3-b3|).

There are no sign constraints on the parameters; in particular, negative arguments are allowed.

The new tetrahedra will be inserted at the end of the list of tetrahedra in the triangulation.

Precondition
gcd(a1, b1) = 1.
gcd(a2, b2) = 1.
gcd(a3, b3) = 1.
Parameters
a1a parameter describing the first layered solid torus in the augmented triangular solid torus.
b1a parameter describing the first layered solid torus in the augmented triangular solid torus.
a2a parameter describing the second layered solid torus in the augmented triangular solid torus.
b2a parameter describing the second layered solid torus in the augmented triangular solid torus.
a3a parameter describing the third layered solid torus in the augmented triangular solid torus.
b3a parameter describing the third layered solid torus in the augmented triangular solid torus.

◆ 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) x S^1.

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

Returns
the product B^(dim-1) x S^1.

◆ bingsHouse()

static Triangulation< 3 > regina::Example< 3 >::bingsHouse ( )
static

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

Returns
a 3-sphere triangulation dual to Bing's house.

◆ cuspedGenusTwoTorus()

static Triangulation< 3 > regina::Example< 3 >::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
the solid genus two torus with cusped boundary.

◆ 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 x 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 x I we "remove" the vertices at the apex of each cone.

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.

◆ figureEight()

static Triangulation< 3 > regina::Example< 3 >::figureEight ( )
static

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

Returns
the figure eight knot complement.

◆ gieseking()

static Triangulation< 3 > regina::Example< 3 >::gieseking ( )
static

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

Returns
the Gieseking manifold.

◆ layeredLoop()

static Triangulation< 3 > regina::Example< 3 >::layeredLoop ( size_t  length,
bool  twisted 
)
static

Returns a layered loop of the given length.

Layered loops are described in detail in the LayeredLoop class notes.

Parameters
lengththe length of the layered loop to construct; this must be strictly positive.
twistedtrue if the layered loop should be twisted, or false if it should be untwisted.
Returns
the resulting layered loop.

◆ lens()

static Triangulation< 3 > regina::Example< 3 >::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. A layered lens space is constructed by building a layered solid torus and then joining together the two boundary triangles.

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
the lens space L(p,q).

◆ lst()

static Triangulation< 3 > regina::Example< 3 >::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
the layered solid torus LST(a,b,c).

◆ poincare()

static Triangulation< 3 > regina::Example< 3 >::poincare ( )
static

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

Returns
the Poincare homology sphere.

◆ poincareHomologySphere()

Triangulation< 3 > regina::Example< 3 >::poincareHomologySphere ( )
inlinestatic

Deprecated routine that returns the five-tetrahedron triangulation of the Poincare homology sphere.

Deprecated:
This routine has been renamed to poincare().
Returns
the Poincare homology sphere.

◆ rp2xs1()

static Triangulation< 3 > regina::Example< 3 >::rp2xs1 ( )
static

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

Returns
the product space RP^2 x S^1.

◆ rp3rp3()

static Triangulation< 3 > regina::Example< 3 >::rp3rp3 ( )
static

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

Returns
the connected sum RP^3 # RP^3.

◆ s2xs1()

Triangulation< 3 > regina::Example< 3 >::s2xs1 ( )
inlinestatic

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

This is identical to calling the generic routine sphereBundle().

Returns
the product space S^2 x S^1.

◆ sfsOverSphere()

static Triangulation< 3 > regina::Example< 3 >::sfsOverSphere ( long  a1 = 1,
long  b1 = 0,
long  a2 = 1,
long  b2 = 0,
long  a3 = 1,
long  b3 = 0 
)
static

Returns a triangulation of the given orientable Seifert fibred space over the sphere with at most three exceptional fibres.

The Seifert fibred space will be SFS((a1, b1), (a2, b2), (a3, b3)), where the parameters a1, ..., b3 are passed as arguments to this routine.

The three pairs of parameters (a, b) do not need to be normalised, i.e., the parameters can be positive or negative and b may lie outside the range [0..a). There is no separate twisting parameter; each additional twist can be incorporated into the existing parameters by replacing some pair (a>, b) with the pair (a, a + b). For Seifert fibred spaces with less than three exceptional fibres, some or all of the parameter pairs may be (1, k) or even (1, 0).

If you wish to construct more complex Seifert fibred spaces (e.g., with more exceptional fibres, or with a different base orbifold), you can use the more sophisticated SFSpace::construct().

Precondition
None of a1, a2 or a3 are 0.
gcd(a1, b1) = 1.
gcd(a2, b2) = 1.
gcd(a3, b3) = 1.
Parameters
a1a parameter describing the first exceptional fibre.
b1a parameter describing the first exceptional fibre.
a2a parameter describing the second exceptional fibre.
b2a parameter describing the second exceptional fibre.
a3a parameter describing the third exceptional fibre.
b3a parameter describing the third exceptional fibre.
Returns
the triangulated Seifert fibred space.

◆ 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.

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 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 (dim-1)-dimensional triangulation base.

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.

◆ smallClosedNonOrblHyperbolic()

static Triangulation< 3 > regina::Example< 3 >::smallClosedNonOrblHyperbolic ( )
static

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

Returns
the closed non-orientable hyperbolic manifold described above.

◆ smallClosedOrblHyperbolic()

static Triangulation< 3 > regina::Example< 3 >::smallClosedOrblHyperbolic ( )
static

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

Returns
the closed orientable hyperbolic manifold described above.

◆ solidKleinBottle()

Triangulation< 3 > regina::Example< 3 >::solidKleinBottle ( )
inlinestatic

Returns a triangulation of the solid Klein bottle.

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

Returns
the solid Klein bottle.

◆ sphere()

Triangulation< dim > regina::detail::ExampleBase< dim >::sphere
staticinherited

Closed Triangulations.

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

Returns
a two-simplex dim-sphere.

◆ sphere600()

static Triangulation< 3 > regina::Example< 3 >::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
the boundary of the regular 600-cell.

◆ sphereBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::sphereBundle
staticinherited

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

Returns
the product S^(dim-1) x S^1.

◆ threeSphere()

Triangulation< 3 > regina::Example< 3 >::threeSphere ( )
inlinestatic

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

This is different from the generic routine sphere(), which uses two tetrahedra instead.

Returns
a one-tetrahedron 3-sphere.

◆ trefoil()

static Triangulation< 3 > regina::Example< 3 >::trefoil ( )
static

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

Returns
the trefoil knot complement.

◆ twistedBallBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::twistedBallBundle
staticinherited

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

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

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

◆ twistedSphereBundle()

Triangulation< dim > regina::detail::ExampleBase< dim >::twistedSphereBundle
staticinherited

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

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

◆ weberSeifert()

static Triangulation< 3 > regina::Example< 3 >::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
the Weber-Seifert dodecahedral space.

◆ weeks()

static Triangulation< 3 > regina::Example< 3 >::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
the Weeks manifold.

◆ whiteheadLink()

static Triangulation< 3 > regina::Example< 3 >::whiteheadLink ( )
static

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

Returns
the Whitehead link complement.

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).