Observed periodicity related to the four-strand Burau representation

TL;DR

This paper investigates the faithfulness of the Burau representation for B_4, using computational methods to show no counterexamples exist with up to 2000 intersections, and discovers a notable periodicity in related polynomials.

Contribution

It provides the first computational certification of the faithfulness of Psi_4 up to 2000 intersections and uncovers a new periodicity pattern in the associated polynomial set.

Findings

No arc-pair with ≤2000 intersections causes unfaithfulness in Psi_4

Discovered a striking periodicity in the set of arc-pair polynomials

First analysis revealing deeper structure in these polynomials

Abstract

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.