Census of Mixed-Platonic Manifolds

In this talk, we will describe the census of mixed-platonic 3-manifolds, extending earlier work on tetrahedral and platonic manifolds by Goerner et al. We will introduce 3-manifold censuses and computational tools for 3-manifolds like Regina and SnapPy. Next, we will introduce the hyperbolic geometry behind these constructions. Specifically, we will discuss polyhedral decompositions of hyperbolic 3-manifolds by regular hyperbolic polyhedra. We will motivate this census by describing some known examples of (mixed-)platonic manifolds that arise from knot theory. We will describe Goerner’s algorithm based on Burton’s isomorphism signatures, a combinatorial triangulation invariant which provides an efficient criterion for pruning branches in the tree of candidate triangulations. Further applications include censuses of closed mixed-platonic manifolds and those with totally geodesic boundary.