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

Authors: Clément Maria and Owen Rouillé

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 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.

Cite as

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)

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  author =	{Maria, Cl\'{e}ment and Rouill\'{e}, Owen},
  title =	{{Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{78:1--78:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.78},
  URN =		{urn:nbn:de:0030-drops-170168},
  doi =		{10.4230/LIPIcs.ESA.2022.78},
  annote =	{Keywords: knots and 3-manifolds, triangulation, hyperbolic structure, Thurston equations}
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