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

Authors Clément Maria , Owen Rouillé

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Clément Maria
  • Inria Sophia Antipolis-Méditerranée, France
Owen Rouillé
  • Inria Sophia Antipolis-Méditerranée, France

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Clément Maria and Owen Rouillé. Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 78:1-78:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • knots and 3-manifolds
  • triangulation
  • hyperbolic structure
  • Thurston equations


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