Lawrence-Krammer-Bigelow representations and dual Garside length of braids

TL;DR

This paper proves that the span of the variable q in the Lawrence-Krammer-Bigelow representation equals twice the dual Garside length of a braid, confirming a conjecture by Krammer using a geometric approach.

Contribution

It establishes a precise relationship between the algebraic representation span and the dual Garside length, providing a new geometric proof of Krammer's conjecture.

Findings

The span of q in the representation equals twice the dual Garside length.

Dual Garside length can be read from a labeling of the curve diagram.

The proof uses a geometric approach similar to Bigelow's.

Abstract

We show that the span of the variable $q$ in the Lawrence-Krammer-Bigelow representation matrix of a braid is equal to the twice of the dual Garside length of the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow's geometric approach. The key observation is that the dual Garside length of a braid can be read off a certain labeling of its curve diagram.