Computing complete hyperbolic structures on cusped 3-manifolds

TL;DR

This paper introduces a new convex optimization-based method to compute complete hyperbolic structures on cusped 3-manifolds, improving triangulation solutions for geometric analysis.

Contribution

It presents a novel approach combining convex optimization and combinatorial modifications to find suitable triangulations for hyperbolic structure computation.

Findings

Successfully computes hyperbolic structures on various 3-manifolds.

Demonstrates the effectiveness of the new method through experimental results.

Provides a systematic way to modify triangulations for better solutions.

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.