Regular Tessellation Link Complements

TL;DR

This paper classifies regular tessellations of hyperbolic 3-manifolds by ideal Platonic solids, identifying all link complements with such tessellations and completing the classification for specific cases, with some cases still unresolved.

Contribution

It provides a complete classification of regular tessellations with small cusp modulus and characterizes infinite volume cases, advancing understanding of hyperbolic link complements.

Findings

At least 19 and at most 21 regular tessellation link complements exist.

Complete classification for principal congruence link complements with discriminants D=-3 and D=-4.

Identification of infinite volume tessellations with large cusp modulus.

Abstract

By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we prove that a regular tessellations has to be 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 likely 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 in Baker and Reid for the cases of discriminant D=-3 and D=-4. We only describe the manifolds arising as complements of links here with a future publication "Regular Tessellation Links" giving explicit pictures of these links.