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In 1997, Cooper-Long established the nontriviality of the kernel of themodulo 2 Burau representation for the 4-strand braid group, $B_4$. This paperextends their work by embedding the image group into a finitely presentedgroup. 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 extensionof $B_4$.

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