Martin David Palmer -- Publication -- The Burau representations of loop braid groups
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The Burau representations of loop braid groups
with Arthur Soulié

Comptes Rendus. Mathématique vol. 360 (2022) pp. 781-797
arXiv: abspdf


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.

Supplementary material

Explicit computations (using Sage) verifying that, for n=4, the matrices given in Table 1 of our paper do indeed satisfy all of the relations of the extended loop braid group — pdf.