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

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 $B_4$.