Abstract
By a regular tessellation, we mean a hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant that we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we prove that a regular tessellation has to be of infinite volume if its fundamental group is generated by peripheral curves only. This shows that there are at least 19 and at most 21 link complements that are regular tessellations (computer experiments suggest that at least one of the two remaining cases fails to be a link complement, but so far, we have no proof). In particular, we complete the classification of all principal congruence link complements given by Baker and Reid for the cases of discriminant D = −3 and D = −4. We describe only the manifolds arising as complements of links here, with a future publication “Regular Tessellation Links” giving explicit pictures of these links.
Notes
1See “Is the 4x5 chessboard complex a link complement?” at http://mathoverflow.net/questions/36791.
4The source code and all other files necessary for the reader easily to certify the correctness of the results in this paper are available at http://www.unhyperbolic.org/regTess/.
5SnapPy is available at http://snappy.computop.org/. Regina can be found at http://regina.sourceforge.net/.
6Gap can be found at http://www.gap-system.org/.
7This technique was suggested at http://mathoverflow.net/questions/159217.
8Called infiniteUniversalRegularTessellationProofs.g.
9Available at http://www.math.chs.nihon-u.ac.jp/~ichihara/ExcAlt/.
10Note that (1, 0) is considered a trivial slope and is not reported by fef_gen.py.