Computing complete hyperbolic structures on cusped 3-manifolds
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.
Cite
@article{arxiv.2112.06360,
title = {Computing complete hyperbolic structures on cusped 3-manifolds},
author = {Clément Maria and Owen Rouillé},
journal= {arXiv preprint arXiv:2112.06360},
year = {2022}
}