Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds
Résumé
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 localized 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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