Triangulations

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

Regina offers a selection of ready-made sample triangulations, which you can play with to discover how Regina works. These include examples of knot complements and 2-knot complements, well-known spaces such as the Poincaré homology sphere and the Weber-Seifert dodecahedral space, and many others. Simply select an example from the list provided and Regina will build the corresponding triangulation for you.

Regina can 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 simplices and their vertices). An example for 3-manifolds 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]. They are available in all of the dimensions that Regina supports, and for 3-manifolds the format is explicitly described in [Bur11c].

If you already have a triangulation and you wish to view its isomorphism signature: for 3-manifolds you can view it through the triangulation composition tab, and in other dimensions you can access it through Python scripting or C++ programming.

Regina can also rehydrate a 3-manifold triangulation from a 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 3-manifold triangulations have dehydration strings. The dehydration string (if it exists) for an existing triangulation can be accessed through Python scripting by calling Triangulation3.dehydrate().

Regina can reconstruct a 3-manifold 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].

For 2-manifolds, Regina can triangulate a connected surface of any topological type. Select either Orientable surface or Non-orientable surface as the type of triangulation, and then enter the genus and the number of punctures that you would like in the surface.

For orientable surfaces the genus represents the number of handles (e.g., the torus has orientable genus 1), and for non-orientable surfaces the genus represents the number of crosscaps (e.g., the Klein bottle has non-orientable genus 2).

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).

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) ... (an,bn), where each pair (ai,bi) 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, ..., an may be zero.

Each pair (ai,bi) does not need to be normalised; that is, 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 (ai,ai+bi). Pairs of the form (1,k) and even (1,0) are acceptable.

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.

This will create an orientable handlebody of the given genus. This is built by layering tetrahedra onto an appropriate one-vertex once-punctured non-orientable surface (thus generalisating the layered solid torus construction, which essentially layers tetrahedra onto a one-vertex Möbius band).

The genus may be any non-negative integer.

This allows you to build a 4-manifold from an existing 3-manifold triangulation, by making either an I-bundle, an S¹-bundle, or a cone.

Choose one of I-bundle over…, S¹-bundle over… or Cone over… as the triangulation type, and then select your 3-manifold triangulation in the Source 3-manifold drop-down box. If your triangulation represents the 3-manifold M, then Regina will build either the 4-manifold M × I, M × S¹, or the cone over M respectively.

You can also build new triangulations from old, using various doubling constructions. To do this, you need to have the source triangulation open for viewing. The doubling constructions will then be available through the Triangulation menu.

These doubling constructions will not change the source triangulation. Instead the result will appear as a new triangulation beneath the original in the packet tree.

In general, these doubling constructions do not give oriented triangulations.

Any simplex and/or facet locks in the source triangulation will not prevent you from making doubling constructions; instead any locks will be copied across to the appropriate locations in the new triangulation. So, for example, in the orientable double cover each lock will be duplicated in both sheets, and if you double over the boundary then any locked boundary facets will become locked internal facets of the new triangulation.

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.

Regina can build a census of all triangulations satisfying a variety of different constraints. The best way to do this is through the command-line tool tricensus.