A census of hyperbolic platonic manifolds and augmented knotted trivalent graphs

TL;DR

This paper provides a comprehensive census of hyperbolic Platonic 3-manifolds, classifies their tessellations, and links some to augmented knotted trivalent graphs, enhancing computational tools like SnapPy.

Contribution

It extends previous work by cataloging all hyperbolic Platonic manifolds and their tessellations, including new classifications for octahedral cases and algorithmic improvements.

Findings

Census of hyperbolic Platonic manifolds and tessellations

Identification of octahedral manifolds as augmented knotted graph complements

Implementation of census data into SnapPy and Regina formats

Abstract

We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedral case, we also identify which manifolds are complements of an augmented knotted trivalent graph and give the corresponding link. A (small version of) the Platonic census and the related improved algorithms have been incorporated into SnapPy. The census also comes in Regina format. In the appendix, we show that an ideal cubical tessellation can be subdivided into an ideal geometric triangulation.