Observed Periodicity Related to the Four-Strand Burau Representation

ABSTRACT

A long-standing open problem is to determine for which values of n the Burau representation Ψ_n of the braid group B_n is faithful. Following the work of Moody, Long–Paton, and Bigelow, the remaining open case is n = 4. One criterion states that Ψ_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 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.

KEYWORDS:

• braid group
• Burau representation
• geometric group theory

2010 AMS SUBJECT CLASSIFICATION:

• 20F36
• 20G05
• 20F65

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.