$L^2$-Burau maps and $L^2$-Alexander torsions

TL;DR

This paper introduces $L^2$-Burau maps to compute $L^2$-Alexander torsions, demonstrating they distinguish more braids than the classical Burau representation, thus advancing link invariants and braid classification.

Contribution

It defines $L^2$-Burau maps and applies them to compute $L^2$-Alexander torsions, showing they outperform the classical Burau representation in distinguishing braids.

Findings

$L^2$-Burau maps can compute $L^2$-Alexander torsions.

$L^2$-Burau maps distinguish more braids than the classical Burau representation.

The approach enhances braid and link invariants.

Abstract

It is well known that the Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define $L^2$-Burau maps and use them to compute some $L^2$-Alexander torsions of links. As an application, we prove that the $L^2$-Burau maps distinguish more braids than the Burau representation.