Computing complete hyperbolic structures on cusped 3-manifolds

Clément Maria; Owen Rouillé

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arXiv · 2112.06360

Computing complete hyperbolic structures on cusped 3-manifolds

Clément Maria·Owen Rouillé

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Abstract

A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic 3-manifold with torus boundaries. This family of 3-manifolds includes the knot complements. This computation of a hyperbolic structure requires the resolution of gluing equations on a triangulation of the space, but not all triangulations admit a solution to the equations. In this paper, we propose a new method to find a triangulation that admits a solution to the gluing equations, using convex optimization and combinatorial modifications. It is based on Casson and Rivin s reformulation of the equations. We provide a novel approach to modify a triangulation and update its geometry, along with experimental results to support the new method.

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BibTeX RIS

Clément Maria, Owen Rouillé. 2021-12-13. Computing complete hyperbolic structures on cusped 3-manifolds. https://arxiv.org/abs/2112.06360

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