Braids with as many full twists as strands realize the braid index

TL;DR

This paper links the fractional Dehn twist coefficient of braids to the Upsilon function, proving that certain highly twisted braids achieve their closure's braid index, confirming a conjecture and exploring related properties.

Contribution

It introduces a new characterization of the fractional Dehn twist coefficient via the Upsilon function and proves that n-twisted braids realize their closure's braid index, confirming a conjecture.

Findings

Braids with fractional Dehn twist coefficient > n-1 realize their closure's braid index.

n-twisted braids realize the braid index of their closure.

Examples provided address optimality of the results.

Abstract

We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function-valued concordance homomorphism defined by Ozsv\'ath, Stipsicz, and Szab\'o. We use this characterization to prove that $n$-braids with fractional Dehn twist coefficient larger than $n-1$ realize the braid index of their closure. As a consequence, we are able to prove a conjecture of Malyutin and Netsvetaev stating that $n$-times twisted braids realize the braid index of their closure. We provide examples that address the optimality of our results. The paper ends with an appendix about the homogenization of knot concordance homomorphisms.