Abstract

The faithfulness of the Burau representation of the 4-strand braid group, $B_4$, remains an open question.
In this work, there are two main results. First, we specialize the indeterminate $t$ to a complex number on the unit circle, and we find a necessary condition for a word of $B_4$ to belong to the kernel of the representation. Second, by using a simple algorithm,
we will be able to exclude a family of words in the generators from belonging to the kernel of the reduced Burau representation.

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