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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
- Benjamin A. Burton. The Next 350 Million Knots. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry (SoCG 2020), volume 164 of Leibniz International Proceedings in Informatics (LIPIcs), pages 25:1-25:17, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.25.
-
Benjamin A. Burton, Ryan Budney, William Pettersson, et al. Regina: Software for low-dimensional topology. http:// regina-normal. github. io/, 1999-2021.
- Marc Culler, Nathan M. Dunfield, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds. https://snappy.math.uic.edu/, 1991-2021.
- David Futer and François Guéritaud. From angled triangulations to hyperbolic structures, 2010. URL: https://doi.org/10.1090/conm/541/10683.
- Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger. The computational complexity of knot and link problems. J. ACM, 46(2):185-211, 1999. URL: https://doi.org/10.1145/301970.301971.
-
William H. Jaco and J. Hyam Rubinstein. 0-efficient triangulations of 3-manifolds. Journal of Differential Geometry, 65:61-168, 2002.
-
D. Kraft. A Software Package for Sequential Quadratic Programming. Deutsche Forschungs- und Versuchsanstalt für Luft- und Raumfahrt Köln: Forschungsbericht. Wiss. Berichtswesen d. DFVLR, 1988.
- Marc Lackenby. Word hyperbolic dehn surgery. Inventiones mathematicae, pages 243-282, 2000. URL: https://doi.org/10.1007/s002220000047.
- Clément Maria and Owen Rouillé. Computing complete hyperbolic structures on cusped 3-manifolds, 2021. URL: https://doi.org/10.48550/ARXIV.2112.06360.
- George D. Mostow. Strong Rigidity of Locally Symmetric Spaces. Princeton University Press, 1973. URL: https://doi.org/doi:10.1515/9781400881833.
- Hitoshi Murakami and Yoshiyuki Yokota. Volume Conjecture for Knots. Springer, 2018. URL: https://doi.org/10.1007/978-981-13-1150-5.
- Barbara Nimershiem. Geometric triangulations of a family of hyperbolic 3-braids, 2021. URL: http://arxiv.org/abs/2108.09349.
- Udo Pachner. P.l. homeomorphic manifolds are equivalent by elementary shellings. European Journal of Combinatorics, 12(2):129-145, 1991. URL: https://doi.org/10.1016/S0195-6698(13)80080-7.
-
Igor Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume. Annals of Mathematics, 139:553-580, 1994.
- W. P. Thurston. The geometry and topology of 3-manifolds, volume 1. Princeton University Press, Princeton, N.J., 1980. Electronic version 1.1 - March 2002. URL: http://library.msri.org/books/gt3m/.
- Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261-272, 2020. URL: https://doi.org/10.1038/s41592-019-0686-2.
- Jeffrey Weeks. Computation of hyperbolic structures in knot theory. Handbook of Knot Theory, October 2003. URL: https://doi.org/10.1016/B978-044451452-3/50011-3.