Syllable Permutations and Hyperbolic Lorenz Knots

Authors

Paulo Gomes, ´Area Departamental de Matem´atica, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Em´ıdio Navarro, 1, 1959-007 Lisboa, PortugalFollow
Nuno Franco, Departamento de Matem´atica, Universidade de ´Evora, Largo dos Colegiais 2, 7004-516 ´Evora, PortugalFollow
Lu?s Silva, ´Area Departamental de Matem´atica, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Em´ıdio Navarro, 1, 1959-007 Lisboa, PortugalFollow

Author Country (or Countries)

Portugal

Abstract

Lorenz knots are the knots corresponding to periodic orbits in the flow associated to the Lorenz system. This flow induces an iterated one-dimensional first-return map whose orbits can be represented, using symbolic dynamics, by finite words. As a result of Thurston’s geometrization theorem, all knots can be classified as either torus, satellite or hyperbolic knots. Birman andWilliams proved that all torus knots are Lorenz knots which can be represented by a class of words with a precise form. We consider about 20000 words corresponding to all non-trivial permutations of a sample of words associated to torus knots and, using the Topology and Geometry software SnapPy, we compute their hyperbolic volume, concluding that it is significantly different from zero, meaning that all these knots are hyperbolic. This leads us to conjecture that all knots in this family are hyperbolic.

Recommended Citation

Gomes, Paulo; Franco, Nuno; and Silva, Lu?s (2023) "Syllable Permutations and Hyperbolic Lorenz Knots," Applied Mathematics & Information Sciences: Vol. 09: Iss. 5, Article 15.
Available at: https://digitalcommons.aaru.edu.jo/amis/vol09/iss5/15