The Burau representations of loop braid groups

TL;DR

This paper presents a topological construction of the Burau representations for loop braid groups, revealing four versions with one notably subtle and topologically natural representation.

Contribution

It introduces a new topological approach to construct all four versions of the Burau representations for loop braid groups, including a previously less obvious one.

Findings

Four versions of Burau representations are constructed

One representation is topologically natural and subtle

The constructions clarify the structure of loop braid groups

Abstract

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural.