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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.
The simplest way to create a triangulation is through the → 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.
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.
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
,