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

#include <triangulation/example3.h>

Inheritance diagram for regina::Example< 3 >:

Static Public Member Functions
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.

Closed Triangulations
static Triangulation< 3 >	threeSphere ()
	Returns a one-tetrahedron triangulation of the 3-sphere.

static Triangulation< 3 >	bingsHouse ()
	Returns the two-tetrahedron triangulation of the 3-sphere that is dual to Bing's house with two rooms.

static Triangulation< 3 >	s2xs1 ()
	Returns a two-tetrahedron triangulation of the product space `S² × S¹`.

static Triangulation< 3 >	rp2xs1 ()
	Returns a three-tetrahedron triangulation of the non-orientable product space `RP² × S¹`.

static Triangulation< 3 >	threeTorus ()
	Returns a six-tetrahedron triangulation of the 3-torus; that is, the product space `S¹ × S¹ × S¹`.

static Triangulation< 3 >	rp3rp3 ()
	Returns a triangulation of the connected sum `RP³ # RP³`.

static Triangulation< 3 >	lens (size_t p, size_t q)
	Returns a triangulation of the lens space `L(p,q)`.

static Triangulation< 3 >	layeredLoop (size_t length, bool twisted)
	Returns a layered loop of the given length.

static Triangulation< 3 >	poincare ()
	Returns the five-tetrahedron triangulation of the Poincare homology sphere.

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.

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.

static Triangulation< 3 >	weeks ()
	Returns a nine-tetrahedron minimal triangulation of the Weeks manifold.

static Triangulation< 3 >	weberSeifert ()
	Returns a one-vertex triangulation of the Weber-Seifert dodecahedral space.

static Triangulation< 3 >	smallClosedOrblHyperbolic ()
	Returns the nine-tetrahedron closed orientable hyperbolic 3-manifold with volume 0.94270736.

static Triangulation< 3 >	smallClosedNonOrblHyperbolic ()
	Returns the eleven-tetrahedron closed non-orientable hyperbolic 3-manifold with volume 2.02988321.

static Triangulation< 3 >	sphere600 ()
	Returns the boundary 3-sphere of the regular 600-cell.

Finite Bounded Triangulations
static Triangulation< 3 >	lst (size_t a, size_t b)
	Returns the layered solid torus `LST(a,b,c)`.

static Triangulation< 3 >	handlebody (size_t genus)
	Returns a triangulation of the orientable handlebody with the given genus.

static Triangulation< 3 >	solidKleinBottle ()
	Returns a triangulation of the solid Klein bottle.

static Triangulation< 3 >	b5 ()
	Returns a triangulation of the Martelli-Petronio brick B5.

Ideal Triangulations
static Triangulation< 3 >	figureEight ()
	Returns a two-tetrahedron ideal triangulation of the figure eight knot complement.

static Triangulation< 3 >	trefoil ()
	Returns a two-tetrahedron ideal triangulation of the trefoil knot complement.

static Triangulation< 3 >	whitehead ()
	Returns a four-tetrahedron ideal triangulation of the Whitehead link complement.

static Triangulation< 3 >	whiteheadLink ()
	Deprecated alias for whitehead(), which returns a four-tetrahedron ideal triangulation of the Whitehead link complement.

static Triangulation< 3 >	gieseking ()
	Returns the one-tetrahedron ideal triangulation of the non-orientable Gieseking manifold.

static Triangulation< 3 >	idealGenusTwoHandlebody ()
	Returns a triangulation of a solid genus two handlebody with ideal boundary.

static Triangulation< 3 >	cuspedGenusTwoTorus ()
	Deprecated routine that returns a triangulation of a solid genus two handlebody with ideal boundary.

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 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 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.

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.

Precondition: gcd(a1, b1) = gcd(a2, b2) = gcd(a3, b3) = 1.

Exceptions

InvalidArgument The preconditions above do not hold; that is, at least one of the pairs (a1, b1), (a2, b2) or (a3, b3) is not coprime.

Parameters

a1	a parameter describing the first layered solid torus in the augmented triangular solid torus.
b1	a parameter describing the first layered solid torus in the augmented triangular solid torus.
a2	a parameter describing the second layered solid torus in the augmented triangular solid torus.
b2	a parameter describing the second layered solid torus in the augmented triangular solid torus.
a3	a parameter describing the third layered solid torus in the augmented triangular solid torus.
b3	a parameter describing the third layered solid torus in the augmented triangular solid torus.

◆ b5()

static Triangulation< 3 > regina::Example< 3 >::b5 ( )

static

Returns a triangulation of the Martelli-Petronio brick B5.

This is an 8-tetrahedron triangulation of a Seifert fibred space with torus boundary. What makes B5 interesting is that, if we keep its boundary fixed, it uses one fewer tetrahedra than the usual prism-and-layered-solid-torus construction for this same Seifert fibred space.

The manifold itself is SFS [D: (2,1) (3,-2)], also known as the trefoil complement.

For more details on the brick B5, see "Complexity of geometric 3-manifolds", Bruno Martelli and Carlo Petronio, Geometriae Dedicata 108 (2004), pp. 15-69.

Returns: the Martell-Petronio brick B5.

◆ 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.

◆ 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()

Triangulation< 3 > regina::Example< 3 >::cuspedGenusTwoTorus ( )

inlinestatic

Deprecated routine that returns a triangulation of a solid genus two handlebody with ideal boundary.

Deprecated: This has been renamed to idealGenusTwoHandlebody(), which is a little less loose with language.

See idealGenusTwoHandlebody() for further details.

Returns: the solid genus two handlebody with ideal 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 × 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.

◆ 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.

◆ handlebody()

static Triangulation< 3 > regina::Example< 3 >::handlebody ( size_t genus )

static

Returns a triangulation of the orientable handlebody with the given genus.

For positive genus, this routine uses a minimal layered triangulation of the orientable handlebody. This is constructed by starting with a one-vertex triangulation of a once-punctured non-orientable surface with the given genus, and layering a tetrahedron onto each internal edge of this surface, yielding a (3*genus-2)-tetrahedron triangulation. For genus greater than one, there are many choices for how to do this; this routine makes an arbitrary choice.

For genus 0, this routine uses the one-tetrahedron 3-ball.

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.

Parameters

genus the genus of the handlebody.

Returns: the orientable handlebody with the given genus.

Author: Alex He

◆ idealGenusTwoHandlebody()

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

static

Returns a triangulation of a solid genus two handlebody with ideal boundary.

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

Prior to Regina 7.4, this routine was called cuspedGenusTwoTorus().

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 solid genus two handlebody with ideal boundary.

◆ 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

length	the length of the layered loop to construct; this must be strictly positive.
twisted	`true` 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.

Exceptions

InvalidArgument The preconditions above do not hold; that is, either q ≥ p and (p,q) ≠ (0,1), and/or p and q are not coprime.

Parameters

p	a parameter of the desired lens space.
q	a 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).

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.

Precondition: gcd(a, b) = 1.

Exceptions

InvalidArgument The preconditions above do not hold; that is, a and b are not coprime.

Parameters

a	the first parameter of the layered solid torus.
b	the 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.

◆ rp2xs1()

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

static

Returns a three-tetrahedron triangulation of the non-orientable product space RP² × S¹.

Returns: the product space RP² × S¹.

◆ rp3rp3()

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

static

Returns a triangulation of the connected sum RP³ # RP³.

Returns: the connected sum RP³ # RP³.

◆ s2xs1()

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

inlinestatic

Returns a two-tetrahedron triangulation of the product space S² × S¹.

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

Returns: the product space S² × S¹.

◆ 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().

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.

Precondition: None of a1, a2 or a3 are 0.; gcd(a1, b1) = gcd(a2, b2) = gcd(a3, b3) = 1.

Exceptions

InvalidArgument The preconditions above do not hold; that is, at least one of a_1, a_2 or a_3 is zero, and/or at least one of the pairs (a1, b1), (a2, b2) or (a3, b3) is not coprime.

Parameters

a1	a parameter describing the first exceptional fibre.
b1	a parameter describing the first exceptional fibre.
a2	a parameter describing the second exceptional fibre.
b2	a parameter describing the second exceptional fibre.
a3	a parameter describing the third exceptional fibre.
b3	a 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.

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.

◆ 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 identical 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.

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.

◆ 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) × 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.

◆ 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.

◆ threeTorus()

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

static

Returns a six-tetrahedron triangulation of the 3-torus; that is, the product space S¹ × S¹ × S¹.

Returns: the product space S¹ × S¹ × S¹.

◆ 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~ 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.

◆ 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.

◆ whitehead()

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

static

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

Returns: the Whitehead link complement.

◆ whiteheadLink()

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

inlinestatic

Deprecated alias for whitehead(), which returns a four-tetrahedron ideal triangulation of the Whitehead link complement.

Deprecated: This routine has been renamed to whitehead().

Returns: the Whitehead link complement.

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

triangulation/example3.h

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ augTriSolidTorus()

◆ b5()

◆ ball()

◆ ballBundle()

◆ bingsHouse()

◆ cuspedGenusTwoTorus()

◆ doubleCone()

◆ figureEight()

◆ gieseking()

◆ handlebody()

◆ idealGenusTwoHandlebody()

◆ layeredLoop()

◆ lens()

◆ lst()

◆ poincare()

◆ rp2xs1()

◆ rp3rp3()

◆ s2xs1()

◆ sfsOverSphere()

◆ simplicialSphere()

◆ singleCone()

◆ smallClosedNonOrblHyperbolic()

◆ smallClosedOrblHyperbolic()

◆ solidKleinBottle()

◆ sphere()

◆ sphere600()

◆ sphereBundle()

◆ threeSphere()

◆ threeTorus()

◆ trefoil()

◆ twistedBallBundle()

◆ twistedSphereBundle()

◆ weberSeifert()

◆ weeks()

◆ whitehead()

◆ whiteheadLink()