Partial classification of Lorenz knots: Syllable permutations of torus knots words

Abstract

We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots coded by some of these families of words are hyperbolic, by showing that they are neither satellites nor torus knots and making use of Thurston's theorem. Infinite families of hyperbolic Lorenz knots are generated in this way, to our knowledge, for the first time. The techniques used can be generalized to study other families of Lorenz knots.

Farey neighbors and hyperbolic Lorenz knots

“…In [10] we proved the following results: Proposition 1. For each word W in P (p, q), 4 < p < q, distinct from W (p, q), there is at most one torus knot T (p, q ′ ), q ′ < q, with the same braid index and genus as the closure of the braid corresponding to W .…”

“…In spite of the extensive computational tests indicate that all knots corresponding to pairs (X, Y ) * S for any Farey pair (X, Y ) are indeed hyperbolic, the results from [10] only allow us to demonstrate this for the following families.…”

“…In [10], based in the work of El-Rifai, [5], we derived an algorithm to obtain Lorenz satellite braids, together with the corresponding words from symbolic dynamics.…”

“…In [10] we derived an algorithm, based in the work of El-Rifai, [6], to obtain Lorenz satellite braids, together with their corresponding aperiodic words and proved that none of the words in the sets P (p, q) can be obtained through this procedure. Thus concluding the following result.…”

Satellites and Lorenz knots

“…It would be useful to classify all satellite, torus, and hyperbolic Lorenz knots. In terms of hyperbolic Lorenz knots, some work has been done to identify families by Gomes, Franco and Silva [9,10]. However, these results assume Morton's conjecture holds, so the status of this work is unclear.…”