Triangulations

This will create a new triangulation with no tetrahedra at all. This option is best if you have a triangulation you wish to enter in by hand, in which case you can create a new empty triangulation and then manually add tetrahedra and edit face gluings.

This will create a layered lens space with parameters (p, q) where both parameters are non-negative, coprime and p>q (with exceptional case (0, 1) also allowed). These two parameters must be provided before the triangulation can be created.

A layered lens space is a lens space built by taking two layered solid tori with appropriate parameters (described below) and gluing them together along their torus boundaries.

Layered lens spaces are discussed by Jaco and Rubinstein [JR03], [JR06] and others, and are described in detail in Burton's PhD thesis [Bur03] which is available from the Regina website.

This will create an orientable Seifert fibred space over the 2-sphere with any number of exceptional fibres. Regina will choose the simplest construction that it knows based upon the given Seifert fibred space parameters. Examples of constructions that are used include layered lens spaces, layered loops and augmented triangular solid tori, all of which are described elsewhere in this section.

The parameters for the Seifert fibred space must be given as a sequence of pairs of integers (a1,b1) (a2,b2) ... (an,bn); an example might be (2,-1) (3,4) (5,-4), which represents the Poincaré homology sphere. Each pair (ai,bi) represents a single exceptional fibre. The two integers in each pair must be relatively prime, and none of a1, a2, ..., an may be zero.

Each pair of parameters (ai,bi) does not need to be normalised, i.e., the parameters may be positive or negative and bi may lie outside the range [0,ai). There is no separate twisting parameter; each additional twist can be incorporated into the existing parameters by replacing some pair (ai,bi) with the pair (ai,ai+bi). Including pairs of the form (1,k) and even (1,0) is acceptable (in which case a lens space may result).

This will create a new layered solid torus with parameters (a, b, c) where all three parameters are non-negative, coprime and a+b=c. These three parameters must be provided before the triangulation can be created.

A layered solid torus is a solid torus built by taking a one-tetrahedron solid torus and repeatedly adding layers. A new layer is added by taking a new tetrahedron and gluing two of its faces as a square onto the existing torus boundary; the remaining two faces of the new tetrahedron then form the new torus boundary. Note that each new layer can be added in three different ways (corresponding to the three different two-face squares that can be chosen on the torus boundary). The parameters of the layered solid torus represent the ways in which the different layers have been added.

Layered solid tori are discussed by Jaco and Rubinstein [JR03], [JR06] and others, and are described in detail in Burton's PhD thesis [Bur03] which is available from the Regina website.

This will create a layered loop of a given length. This length must be provided before the triangulation can be created. It must also be specified whether the layered loop should be twisted or untwisted.

A layered loop of length n is formed by layering n tetrahedra one upon another to form a closed loop of tetrahedra. In a twisted layered loop, a 180 degree rotation takes place before the final layering is performed.

Layered loops are described in detail in Burton's PhD thesis [Bur03], which is available from the Regina website.

Note that a twisted layered loop of length n forms a one-vertex triangulation of the orbit manifold S³/Q_4n, and an untwisted layered loop of length n forms a two-vertex triangulation of the lens space L(n,1).

This will create an augmented triangular solid torus with the given parameters. An augmented triangular solid torus is created by starting with a core three-tetrahedron solid torus and attaching three layered solid tori to its boundary.

Details of the augmented triangular solid torus construction can be found in Burton's PhD thesis [Bur03], which is available from the Regina website.

Parameters for the augmented triangular solid torus must be provided before the construction can take place; these parameters must be given as three pairs of integers (a1,b1) (a2,b2) (a3,b3). These pairs of integers describe the three layered solid tori that are attached to the core as described above. The two integers in each pair must be relatively prime, and both positive and negative integers are allowed.

Note that if none of a1, a2 or a3 are zero then an augmented triangular solid torus always produces a Seifert fibred space over the sphere with at most three exceptional fibres. Conversely, any Seifert fibred space with these properties can be represented as an augmented triangular solid torus.

This will create the triangulation obtained by rehydrating the given dehydration string.

A dehydration string is a set of letters such as dadbcccaqhx containing enough information to recreate the triangulation. Dehydration strings appear in census papers such as the hyperbolic cusped census of Callahan, Hildebrand and Weeks [CHW99], in which the dehydration format is explicitly described.

The dehydration string for an existing triangulation can be viewed through the triangulation composition tab.

This will create the triangulation containing the given splitting surface.

A splitting surface is a compact normal surface consisting of precisely one quadrilateral per tetrahedron and no other normal (or almost normal) discs.

A splitting surface signature is a string of letters arranged into cycles that represent the quadrilateral structure of a splitting surface. From this string of letters the splitting surface and then the enclosing triangulation can be reconstructed.

When entering a splitting surface signature, any block of punctuation will be assumed to separate cycles of letters. All whitespace will be ignored. Examples of valid signatures are (ab)(bC)(Ca) and AAb-bc-C.

The precise format of splitting surface signatures is described in Burton's PhD thesis [Bur03], which is available from the Regina website.

The New Triangulation dialog includes a small selection of ready-made example triangulations that can be created on the fly; examples include the figure eight knot complement, the Poincaré homology sphere and the Seifert-Weber dodecahedral space, amongst others. Simply select a 3-manifold from the list provided and the corresponding triangulation will be created for you.