L²-Burau maps of braids are generalizations of the Burau representation of braid groups, that A. Conway and I defined in 2018. The coefficients of the L²-Burau matrices may no longer be Laurent polynomials but instead live in a general non abelian group ring. We previously linked some of these L²-Burau maps to L²-Alexander torsions of links, which are powerful but elusive link invariants.

In these two talks, I will go further and provide answers to a natural question: how can one define other link invariants from general L²-Burau maps ? To this end, I studied the influence of Markov moves of braids on L²-Burau maps, and I computed new values of L²-determinants via combinatorics on Cayley graphs of groups.

In the first part of the talk I will recall basic facts about braid groups, link invariants, the Burau representation and L²-invariants.

In the second part I will go into more details on how to construct link invariants from L²-Burau maps.