Census of the 191 orientable octahedral manifolds

Damian Heard, Katya Pervova and I have classified in [49] and [50] the compact orientable 3-manifolds
that one can obtain by gluing together in pairs the faces of the octahedron and then removing open regular
neighbourhoods of the singular points. There are such 191 manifolds, and a complete list of their triangulations
in SnapPea format is provided below. 132 of these manifolds are hyperbolic (with geodesic boundary and/or toric cusps).

There are 298 combinatorially inequivalent orientation-reversing gluing patterns for the faces of the octahedron,
available here. After subdivision of the octahedron into 4 tetrahedra they give rise to 298
triangulations, so our main task was to decide which of them encode homeomorphic manifolds. We provide here the list
of all these 298 triangulations, that one may use to test the effectiveness of a computer program aimed at
distinguishing and matching triangulated 3-manifolds. We provide one separate list for each homeomorphism
type of the boundary:
empty (37 triangulations),
one torus (81 triangulations),
two tori (9 triangulations),
one genus-2 surface (113 triangulations),
one torus and one genus-2 surface (2 triangulations),
one genus-3 surface (56 triangulations; this file has a slightly different format than the other ones, because it comes from a previous census).

The (non-user-friendly) Haskell code using which we have obtained the above data is here.

The matching of the triangulations was performed using Orb and later checked and complemented by theoretical work.
The result is the following list, subdivided according to the topological type of the boundary and hyperbolicity.
The first column refers to one of the triangulation files listed above and the second column is the position
(starting from 0) of the triangulation in this file:

Closed (17 manifolds, all non-hyperbolic),
One-cusped hyperbolic (9 manifolds),
One-cusped non hyperbolic (21 manifolds),
Two-cusped hyperbolic (2 manifolds),
Two-cusped non hyperbolic (5 manifolds),
Genus-2 boundary hyperbolic (63 manifolds),
Genus-2 boundary non hyperbolic (16 manifolds),
One-cusped and genus-2 boundary (2 manifolds, both hyperbolic),
Genus-3 boundary (56 manifolds, all hyperbolic; this file has a slightly different format than the other ones, because it comes from a previous census).

Page last updated on February 29, 2008