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Regina offers a wealth of information about 3-manifold triangulations, spread across the many different tabs in the triangulation viewer. Here we walk through the different properties and invariants that Regina can compute.
At the top of each triangulation viewer is a banner listing some basic properties of the triangulation (circled in red above). The following words might appear:
Signifies that the triangulation has no boundary faces and no ideal vertices. In other words, the link of every vertex is a 2-sphere.
Signifies that at least one vertex of the triangulation is ideal. That is, the vertex link is a closed surface but not a 2-sphere.
You can locate any ideal vertices using the skeleton viewers.
Signifies that the triangulation contains one or more boundary faces.
The words orientable or non-orientable indicate whether or not the triangulation represents an orientable 3-manifold.
If the words orientable and oriented appear, this indicates that the vertex labels 0, 1, 2 and 3 on each tetrahedron induce a consistent orientation for all tetrahedra in the entire triangulation.
If you need a consistent orientation for all tetrahedra but you only see orientable (not orientable and oriented), you can fix this by orienting your triangulation.
The words connected or disconnected indicate whether or not the triangulation forms a single connected piece.
Signifies that the triangulation is “broken” to the point where Regina cannot do any serious work with it. This can happen for one of two reasons: (i) some vertex link is a surface with boundary but not a disc; or (ii) some edge is identified with itself in reverse.
You can locate the offending vertex or edge using the skeleton viewers. If the triangulation is invalid, no other information will appear in the banner.
Signifies that the triangulation contains no tetrahedra at all. In this case, no other information will appear in the banner.
The Gluings tab shows how the various tetrahedron faces are glued to each other in pairs. The face gluings are presented in a table: each row represents a tetrahedron, and the four columns on the right represent the four faces of each tetrahedron. Tetrahedra are numbered 0,1,2,..., and the four vertices of each tetrahedron are numbered 0,1,2,3.
Each cell of this table represents a single face of a single tetrahedron. For instance, the cell circled in red above represents face 123 of tetrahedron 5 (that is, the face formed from vertices 1,2,3 of tetrahedron 5).
The contents of the cell show how the face is glued. In the example
above, the circled cell contains 2 (301)
,
indicating that face 123 of tetrahedron 5 is glued to
face 301 of tetrahedron 2 using the affine map that
matches vertices 1,2,3 of tetrahedron 5 with vertices
3,0,1 of tetrahedron 2 respectively.
The same gluing can be seen from the opposite direction in the row
for tetrahedron 2.
An empty cell indicates that a face is not glued to anything at all; that is, the face forms part of the boundary of the 3-manifold. In the table above there are two boundary faces: face 023 of tetrahedron 2, and face 123 of tetrahedron 4. In our example these join together to form the torus boundary of the figure eight knot complement.
You can modify the triangulation by typing new face gluings directly into this table. See the section on modifying triangulations for details.