A census of Platonic manifolds

Geometry Topology Seminar
Friday, February 5, 2016 - 14:05
1 hour (actually 50 minutes)
Skiles 006
We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Many key examples in 3-manifold topology are Platonic manifolds, e.g., the Poincar\'e homology sphere, the Seifert-Weber dodecahedral space and the complements of the figure eight knot, the Whitehead link, and the minimally twisted 5-component chain link. They have a strong connection to regular tessellations and illustrate many phenomena such as hidden symmetries.I will talk about recent work on a census of hyperbolic Platonic manifolds and some new techniques we developed for its creation, e.g., verified canonical cell decompositions and the isometry signature which is a complete invariant of a cusped hyperbolic manifold.