ABSTRACT
A 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.
Acknowledgments
The authors would like to thank Stephen Bigelow, Dan Margalit, and Balázs Strenner for helpful conversations. Part of this work was completed while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester.