The Lorenz braid index and hyperbolic volume

Abstract

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

28 pages, 4 figures. This is an expanded version of some sections of arXiv:2410.04391v2, the mathematical content of which has been re-organised into two preprints with better focuses and exposition: arXiv:2410.04391v3 and this article. An additional result is included