Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types

TL;DR

This paper investigates the geometric types of Lorenz links by explicitly computing their minimal braids, establishing their relation to V-links, and applying these findings to classify and generalize families of satellite and hyperbolic T-links.

Contribution

It provides explicit minimal braid representations of Lorenz links and relates them to V-links, advancing understanding of their geometric types and classifications.

Findings

Lorenz links are equivalent to V-links via minimal braids.

Explicit formulas relate Lorenz links to V-links.

Conditions on positive braids lead to new classifications of T-links.

Abstract

Lorenz links and T-links are equivalent families of links by Birman and Kofman. Lorenz links are periodic orbits of the Lorenz system. T-links are links given by certain positive braids. Birman, Williams, and Franks proved that every Lorenz link has a positive braid with at least one full twist as a minimal braid. In this paper we study the geometric types of the Lorenz links from their minimal braid perspective. First, we compute their minimal braids explicitly in terms of the parameters of the T-links. We conclude that they are in equivalence with a family of links, called V-links, similar to a result of Birman and Kofman. Furthermore, we find an explicit relation to obtain an element of one of these families from the other. Then, we study conditions on positive braids with at least one full twist to find their geometric type. Finally, we apply these conditions on V-links to find new and generalize some families of satellite and hyperbolic T-links.