# 3-Manifold Triangulations

## Chapter 3. 3-Manifold Triangulations

3-manifolds in Regina are typically represented by triangulations. A triangulation of a 3-manifold consists of a set of tetrahedra with instructions on how some or all of the faces of these tetrahedra should be glued together in pairs.

Triangulations in Regina are less strict than simplicial complexes: you may glue two faces of the same tetrahedron together, or you may glue faces so that different edges of the same tetrahedron become identified (and likewise for vertices). Indeed, the best triangulations for computation are often one-vertex triangulations, where all vertices of all tetrahedra become identified together.

The downside of this flexibility is that, if you are not careful, your triangulation might not represent a 3-manifold at all. If this happens, Regina will tell you about it when you open it to view.

## Creation

### New Triangulations

The simplest way to create a triangulation is through the Packet Tree->New 3-D Triangulation menu item (or the corresponding toolbar button), which will create a new triangulation from scratch.

In addition to the usual information, you are asked what type of triangulation to create (see the drop-down box below). Here we walk through the various options.

#### Empty

This will create a new triangulation with no tetrahedra at all. This is best if you wish to enter a triangulation by hand: first create an empty triangulation, and then manually add tetrahedra and edit the face gluings.

#### Layered Lens Space

This will create a layered lens space with the given parameters. This involves building two layered solid tori and gluing them together along their torus boundaries. Layered lens spaces were introduced by Jaco and Rubinstein [JR03], [JR06] and others.

The parameters (`p`, `q`) must be non-negative and coprime, and must satisfy `p`>`q` (although the exceptional case (0, 1) is also allowed). The resulting 3-manifold will be the lens space L(`p`,`q`).

#### Seifert Fibred Space

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 can based upon the given parameters.

The parameters for the Seifert fibred space must be given as a sequence of pairs of integers (`a1`,`b1`) (`a2`,`b2`) ... (`a`n``,`b`n``), where each pair (`a`i``,`b`i``) describes a single exceptional fibre. An example is (2,-1) (3,4) (5,-4), which represents the Poincaré homology sphere. The two integers in each pair must be relatively prime, and none of `a1`, `a2`, ..., `a`n`` may be zero.

Each pair (`a`i``,`b`i``) does not need to be normalised; that is, the parameters may be positive or negative, and `b`i`` may lie outside the range [0,`a`i``). There is no separate twisting parameter; each additional twist can be incorporated into the existing parameters by replacing some pair (`a`i``,`b`i``) with (`a`i``,`a`i``+`b`i``). Pairs of the form (1,`k`) and even (1,0) are acceptable.

#### Layered Solid Torus

This will create a layered solid torus with the given parameters. This is a solid torus built from a two-triangle Möbius band by repeatedly adding new layers of tetrahedra onto the boundary. Layered solid tori were introduced by Jaco and Rubinstein [JR03], [JR06] and others.

The three parameters (`a`, `b`, `c`) must be non-negative and coprime, and one must be the sum of the other two. These parameters describe how many times the meridional disc of the solid torus intersects the three edges on the boundary of the triangulation.

#### Layered Loop

This will create a layered loop of the given length. This involves layering `n` tetrahedra one upon another (where `n` is the given length), and then gluing the final tetrahedron back around to the first. If the Twisted box is checked, this final gluing will be done with a a 180-degree rotation. Full details of the construction can be found in [Bur03].

A twisted layered loop of length `n` forms a one-vertex triangulation of the orbit manifold S3/Q4`n`. An untwisted layered loop of length `n` forms a two-vertex triangulation of the lens space L(`n`,1).

#### Augmented Triangular Solid Torus

This will create an augmented triangular solid torus with the given parameters. An augmented triangular solid torus is created by building a three-tetrahedron solid torus and then attaching three layered solid tori to its boundary. Details of the construction can be found in [Bur03].

You must provide six parameters, grouped into three pairs of integers (`a1`,`b1`) (`a2`,`b2`) (`a3`,`b3`). Each pair of integers describes one of the layered solid tori that is attached. The two integers in each pair must be relatively prime, and both positive and negative integers are allowed.

If none of `a1`, `a2` or `a3` is zero, the resulting 3-manifold will be a Seifert fibred space over the sphere with at most three exceptional fibres. Conversely, any Seifert fibred space of this type can be represented as an augmented triangular solid torus.

#### Isomorphism Signature

This will reconstruct a triangulation from an isomorphism signature. An isomorphism signature is a compact sequence of letters, digits and/or punctuation that identifies a triangulation uniquely up to combinatorial isomorphism (i.e., relabelling tetrahedra and their vertices). An example is `cPcbbbiht` (which describes the figure eight knot complement).

Stated precisely: every triangulation has a unique isomorphism signature, and two triangulations have the same signature if and only if they are isomorphic. Isomorphism signatures are introduced in the paper [Bur11b], and the format is explicitly described in [Bur11c].

The isomorphism signature for an existing triangulation can be viewed through the triangulation composition tab.

### Caution

Isomorphism signatures are case sensitive! Be sure that you are entering upper-case and lower-case correctly (or better, copy and paste the signature using the clipboard if you can).

#### Dehydration

This will rehydrate a triangulation from the given dehydration string. A dehydration string is a sequence of letters that contains enough information to reconstruct a triangulation (though tetrahedra and their vertices might be relabelled). An example is `dadbcccaqhx` (which describes the SnapPea census triangulation `m025`). 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.

Only some triangulations have dehydration strings. The dehydration string (if it exists) for an existing triangulation can be viewed through the triangulation composition tab.

#### Splitting Surface

This will reconstruct a triangulation from a splitting surface signature. A splitting surface is a compact normal surface consisting of precisely one quadrilateral per tetrahedron and no other normal discs. A splitting surface signature is a string of letters arranged into cycles that describe how these quadrilaterals are joined together. From this signature, both the normal surface and the enclosing triangulation can be reconstructed.

When entering a splitting surface signature, you may use any block of punctuation 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 [Bur03].

#### Example Triangulation

Regina also offers a small selection of ready-made sample triangulations; these include the figure eight knot complement, the Poincaré homology sphere, the Weber-Seifert dodecahedral space, and several others. Simply select one from the list provided and the corresponding triangulation will be built for you.

### Importing Triangulations

You can import triangulations into Regina from other programs, such as SnapPea / SnapPy or Orb. This is done through the File->Import menu. For details, see the chapter on importing and exporting data.

### Creating a Census

Regina can build a census of all 3-manifold triangulations satisfying a variety of different constraints. The best way to do this is through the command-line tool tricensus. For very long calculations, tricensus-mpi may be used to distribute the computation across a cluster of machines.