Characterizing the kernel of the Burau representation modulo 2 for the 4-strand braid group

TL;DR

This paper advances understanding of the 4-strand braid group's Burau representation modulo 2 by embedding its kernel into a finitely presented group and characterizing it via group-theoretic constructions.

Contribution

It extends prior work by embedding the kernel into a finitely presented group and characterizing it as an intersection involving an HNN extension.

Findings

Kernel is nontrivial for B_4 modulo 2

Kernel characterized as intersection with a normal closure

Embedding into a finitely presented group achieved

Abstract

In 1997, Cooper-Long established the nontriviality of the kernel of the modulo 2 Burau representation for the 4-strand braid group, $B_4$. This paper extends their work by embedding the image group into a finitely presented group. As an application, we characterize the kernel as the intersection of $B_4$ with a normal closure of a finite number of elements in an HNN extension of $B_4$.