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.
@InProceedings{maria_et_al:LIPIcs.ESA.2022.78, 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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