Braid groups and euclidean simplices

TL;DR

This paper provides a geometric interpretation of the q parameter in the Lawrence-Krammer-Bigelow representation of braid groups, relating it to actions on Euclidean simplices and revealing elementary reshaping operations.

Contribution

It introduces a novel geometric perspective on the q variable, connecting braid group actions to reshaping Euclidean simplices, expanding understanding of the LKB representation.

Findings

q variable linked to Euclidean simplex reshaping

Braid group actions include relabeling and rescaling simplices

Provides geometric insight into algebraic braid group representations

Abstract

When Daan Krammer and Stephen Bigelow independently proved that braid groups are linear, they used the Lawrence-Krammer-Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid group and plays a major role in Krammer's algebraic proof. The q variable, associated with the dual Garside structure of the braid group, has received less attention. In this article we give a geometric interpretation of the q portion of the LKB representation in terms of an action of the braid group on the space of non-degenerate euclidean simplices. In our interpretation, braid group elements act by systematically reshaping (and relabeling) euclidean simplices. The reshapings associated to the simple elements in the dual Garside structure of the braid group are of an especially elementary type that we call relabeling and rescaling.