Verified computations for closed hyperbolic 3-manifolds

Matthias Goerner

doi:10.1112/blms.12445

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arXiv · 1904.12095

Verified computations for closed hyperbolic 3-manifolds

Matthias Goerner

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Abstract

Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3-manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann-Zagier and Moser for ideal triangulations upon which HIKMOT is based) showing that there is a redundancy among the edge equations if the edges avoid "gimbal lock". We successfully test the algorithm on known examples such as the orientable closed manifolds in the Hodgson-Weeks census and the bundle census by Bell. We also tackle a previously unsolved problem and determine all knots and links with up to 14 crossings that have a hyperbolic branched double cover.

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Matthias Goerner. 2019-04-27. Verified computations for closed hyperbolic 3-manifolds. https://doi.org/10.1112/blms.12445

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