Satellites and Lorenz knots

TL;DR

This paper constructs numerous Lorenz knots that are satellites but not cables, disproving a conjecture and proposing a broader, more accurate conjecture about the structure of Lorenz knots.

Contribution

It introduces new families of Lorenz knots that are satellites but not cables and refines the conjecture regarding their structure and properties.

Findings

Counterexamples to Morton’s conjecture

Broader validity of the amended conjecture

High Dehn filling on parent links produces many Lorenz knots

Abstract

We construct infinitely many families of Lorenz knots that are satellites but not cables, giving counterexamples to a conjecture attributed to Morton. We amend the conjecture to state that Lorenz knots that are satellite have companion a Lorenz knot, and pattern equivalent to a Lorenz knot. We show this amended conjecture holds very broadly: it is true for all Lorenz knots obtained by high Dehn filling on a parent link, and other examples.