Abstract
ABSTRACTA long-standing open problem is to determine for which values of n the Burau representation Ψn of the braid group Bn is faithful. Following the work of Moody, Long–Paton, and Bigelow, the remaining open case is n = 4. One criterion states that Ψn is unfaithful if and only if there exists a pair of arcs in the n-punctured disk Dn such that a certain associated polynomial is zero. In this article, we use a computer search to show that there is no such arc-pair in D4 with 2000 or fewer intersections, thus certifying the faithfulness of Ψ4 up to this point. We also investigate the structure of the set of arc-pair polynomials, observing a striking periodicity that holds between those that are, in some sense, “closest” to zero. This is the first instance known to the authors of a deeper analysis of this polynomial set.
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