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

Authors Clément Maria , Owen Rouillé

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Subject Classification

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

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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 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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ESA.2022.78

Author Details

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

Funding

  • Maria, Clément: Partially funded by ANR project ANR-20-CE48-0007 (AlgoKnot).
  • Rouillé, Owen: Fully funded by ANR project ANR-20-CE48-0007 (AlgoKnot).

References

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