$n$-bridge braids and the braid index

TL;DR

This paper derives a closed-form formula for the braid index of $n$-bridge braids, a broad class of positive braid knots, unifying various knot types and clarifying their definitions.

Contribution

It provides an elementary, effective formula for the braid index of $n$-bridge braids, generalizing previous results and unifying different definitions of twisted torus knots.

Findings

Closed-form formula for braid index of $n$-bridge braids

Unification of twisted torus knot definitions

Partial recovery of Birman--Kofman's work

Abstract

In this work, we find a closed form formula for the braid index of an $n$-bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary, effective, and self-contained, and partially recovers work of Birman--Kofman. Along the way, we show that the disparate definitions of twisted torus knots in the literature agree.