Link invariants from $L^2$-Burau maps of braids

TL;DR

This paper extends the $L^2$-Burau map framework for braids to all quotients of the braid group, producing twisted $L^2$-Alexander torsion invariants that encode rich topological information like hyperbolic volume.

Contribution

It generalizes the $L^2$-Burau map to broader group quotients, linking it to twisted $L^2$-Alexander torsions and analyzing their invariance properties.

Findings

Constructed new link invariants from $L^2$-Burau maps.

Connected $L^2$-Alexander torsions to hyperbolic volume.

Identified limitations of Markov move invariance for these invariants.

Abstract

A previous work of A. Conway and the author introduced $L^2$-Burau maps of braids, which are generalizations of the Burau representation whose coefficients live in a more general group ring than the one of Laurent polynomials. This same work established that the $L^2$-Burau map of a braid at the group of the braid closure yields the $L^2$-Alexander torsion of the braid closure in question, as a variant of the well-known Burau-Alexander formula. In the present paper, we generalize the previous result to $L^2$-Burau maps defined over all quotients of the group of the braid closure. The link invariants we obtain are twisted $L^2$-Alexander torsions of the braid closure, and recover more topological information, such as the hyperbolic volumes of Dehn fillings. The proof needs us to first generalize several fundamental formulas for $L^2$-torsions, which have their own independent interest. We then discuss how likely we are to generalize this process to yet more groups. In particular, a detailed study of the influence of Markov moves on $L^2$-Burau maps and two explicit counter-examples to Markov invariance suggest that twisted $L^2$-Alexander torsions of links are the only link invariants we can hope to build from $L^2$-Burau maps with the present approach.