$n$-bridge braids and the braid index

Dane Gollero, Siddhi Krishna, Marissa Loving, Viridiana Neri, Izah Tahir, Len White

Published 2022-08-29Version 1

Lorenz knots are an important and well studied family of knots. Birman--Kofman showed that Lorenz knots are in correspondence with $T$-links, which are positive braid closures. They determined the braid index of any $T$-link, based on their defining parameters. In this work, we leverage Birman--Kofman's correspondence to show that 1-bridge braids (and, more generally, $n$-bridge braids) are Lorenz knots. Then, we determine the braid index of 1-bridge braids and $n$-bridge braids; our proof is independent of Birman--Kofman's work, and is explicit, elementary, and effective in nature.