The Burau representations 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 nonextended 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.11468
 Bibcode:
 2021arXiv210911468P
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Algebraic Topology;
 20C12;
 20F36;
 20J05;
 57M07;
 57M10
 EPrint:
 15 pages, 11 figures, 1 table. Final version, to appear in Comptes Rendus  Math\'ematique