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Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds
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Clément Maria 
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30th Annual European Symposium on Algorithms (ESA 2022)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: European Symposium on Algorithms (ESA) - License:
Creative Commons Attribution 4.0 International license
- Publication date: 2022-09-01
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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)
https://doi.org/10.4230/LIPIcs.ESA.2022.78
https://doi.org/10.4230/LIPIcs.ESA.2022.78
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.
Subject Classification
ACM Subject Classification
- Theory of computation → Computational geometry
Keywords
- knots and 3-manifolds
- triangulation
- hyperbolic structure
- Thurston equations
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References
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