Regina Calculation Engine
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Offers routines for constructing a variety of sample 3-dimensional triangulations. More...
#include <triangulation/example3.h>
Static Public Member Functions | |
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 > * | poincareHomologySphere () |
Returns the five-tetrahedron triangulation of the Poincare homology sphere. 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... | |
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... | |
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.
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staticinherited |
Bounded Triangulations.
Returns a one-simplex triangulation of the dim-ball.
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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.
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Returns the two-tetrahedron triangulation of the 3-sphere that is dual to Bing's house with two rooms.
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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.
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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.
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Returns a two-tetrahedron ideal triangulation of the figure eight knot complement.
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Returns the one-tetrahedron ideal triangulation of the non-orientable Gieseking manifold.
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Returns a triangulation of the lens space L(p,q)
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The triangulation uses a layered lens space, which is conjectured (but not proven in all cases) to be the triangulation requiring the fewest tetrahedra.
p | a parameter of the desired lens space. |
q | a parameter of the desired lens space. |
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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).
a | the first parameter of the layered solid torus. |
b | the second parameter of the layered solid torus. |
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Returns the five-tetrahedron triangulation of the Poincare homology sphere.
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Returns a three-tetrahedron triangulation of the non-orientable product space RP^2 x S^1
.
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Returns a triangulation of the connected sum RP^3 # RP^3
.
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staticinherited |
Returns the standard (dim+2)-simplex triangulation of the dim-sphere as the boundary of a (dim+1)-simplex.
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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.
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Returns the eleven-tetrahedron closed non-orientable hyperbolic 3-manifold with volume 2.02988321.
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Returns the nine-tetrahedron closed orientable hyperbolic 3-manifold with volume 0.94270736.
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Returns a triangulation of the solid Klein bottle.
This is isomorphic to the triangulation returned by the generic routine twistedBallBundle(), though it will have a different packet label.
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Closed Triangulations.
Returns a two-simplex triangulation of the dim-sphere.
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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
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Returns a two-simplex triangulation of the product space S^(dim-1) x S^1
.
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Returns a two-tetrahedron ideal triangulation of the trefoil knot complement.
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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.
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staticinherited |
Returns a two-simplex triangulation of the twisted product space S^(dim-1) x~ S^1
.
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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.
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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).
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Returns a four-tetrahedron ideal triangulation of the Whitehead link complement.