A new twist on Lorenz links

TL;DR

This paper establishes a correspondence between Lorenz links and T--links, revealing new symmetries, braid index formulas, and identifying many hyperbolic knots as Lorenz knots, with bounded hyperbolic volume and Mahler measure.

Contribution

It introduces a one-to-one correspondence between Lorenz links and T--links, leading to new insights, formulas, and classifications in knot theory.

Findings

Over half of the simplest hyperbolic knots are Lorenz knots.

Hyperbolic volume and Mahler measure are bounded for infinite Lorenz link collections.

New braid index formulas for T--links.

Abstract

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T--links coincide. Using this correspondence, we identify over half of the simplest hyperbolic knots as Lorenz knots. We show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links. The correspondence provides unexpected symmetries for both Lorenz links and T-links, and establishes many new results for T-links, including new braid index formulas.