Mathematics Colloquia and Seminars

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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 Poincare 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. We use a new method to enumerate triangulations with a given property. This method seems to scale better allowing us, for example, to find all hyperbolic manifolds obtained by gluing up to 25 regular ideal tetrahedra. We also found that, like the figure-eight knot complement, the Seifert-Weber space has a sister that cannot be distinguished by its homology.