Partial Classification of Lorenz Knots: Syllable Permutations of Torus Knots Words

TL;DR

This paper introduces a method to generate infinite families of hyperbolic Lorenz knots through syllable permutations of words associated with torus knots, expanding understanding of their classification.

Contribution

It defines a new class of aperiodic words for Lorenz knots and provides an algorithm to construct symbolic words for satellite Lorenz knots, proving hyperbolicity under certain conditions.

Findings

Generated infinite families of hyperbolic Lorenz knots.

Proved some Lorenz knots are hyperbolic using Thurston's theorem.

Established a framework for studying other Lorenz knot families.

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.