  A long-standing open problem is to determine for which values of n the Burau
representation Psi_n of the braid group B_n is faithful. Following work of
Moody, Long-Paton, and Bigelow, the remaining open case is n = 4. One criterion
states that Psi_n is unfaithful if and only if there exists a pair of arcs in
the n-punctured disk D_n such that a certain associated polynomial is zero. In
this paper, we use a computer search to show that there is no such arc-pair in
D_4 with 2000 or fewer intersections, thus certifying the faithfulness of Psi_4
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.
