Syllable Permutations and Hyperbolic Lorenz Knots

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 and Williams 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.