Garside-theoretic analysis of Burau representations

TL;DR

This paper explores the connections between Garside structures of braid groups and the Burau representation, providing criteria for non-vanishing and injectivity in specific cases, advancing understanding of braid group representations.

Contribution

It introduces a non-vanishing criterion for the Burau representation of 4-strand braids and shows injectivity for simply-nested braids in the dual Garside structure.

Findings

Non-vanishing criterion for 4-strand Burau representation

Injectivity of Burau representation for simply-nested braids

Relations between classical and dual Garside structures

Abstract

We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the 4-strand braid group. In the dual context, it is shown that the Burau representation for arbitrary braid index is injective when restricted to the set of \emph{simply-nested braids}.