New York Journal of Mathematics
Volume 23 (2017) 527-553


Matthias Goerner

A census of hyperbolic Platonic manifolds and augmented knotted trivalent graphs

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Published: May 11, 2017
Keywords: Hyperbolic 3-manifolds, regular ideal Platonic solids, census, Platonic manifolds
Subject: Primary 57N10. Secondary 57M25

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.


Partially supported by National Science Foundation grant DMS-11-07452

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