A Census of Tetrahedral Hyperbolic Manifolds

ABSTRACT

We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orientable case) tetrahedra. Our isometry classification uses certified canonical cell decompositions (based on work by Dunfield, Hoffman, and Licata) and isomorphism signatures (an improvement of dehydration sequences by Burton). The tetrahedral census comes in Regina as well as SnapPy format, and we illustrate its features.

KEYWORDS:

• hyperbolic 3-manifolds
• regular ideal tetrahedron
• census
• tetrahedral manifolds
• Bianchi orbifolds

2000 AMS Subject Classification:

• Primary 57N10
• Secondary 57M25

Acknowledgments

We would like to thank Frank Swenton for adding the features to the Kirby calculator [Swenton 14] necessary to draw the new tetrahedral links, and Cameron Gordon for completing the proof of Lemma 6.1.

Funding

E.F., V.T., and A.V. were supported in part by the Ministry of Education and Science of the Russia (the state task number 1.1260.2014/K) and RFBR grant 16-01-00414. S.G. was supported in part by a National Science Foundation grant DMS-14-06419. M.G. was supported in part by a National Science Foundation grant DMS-11-07452.

Notes

1 The case of the letter m is exceptional because it spans several number of tetrahedra for purely historic reasons.

2 We plan a future publication describing how to generalize the techniques for certifying isometry signatures to all cusped hyperbolic manifolds. The third named author has already incorporated this into SnapPy, beginning with version 2.3.2, see SnapPy documentation.