The Burau representation and shapes of polyhedra

TL;DR

This paper links the Burau representation at roots of unity to moduli spaces of Euclidean cone metrics, using orbifold theory to identify kernels and address faithfulness questions for 4-strand braids.

Contribution

It provides a geometric interpretation of the Burau representation at roots of unity and identifies its kernel in the 4-strand case, advancing understanding of its faithfulness.

Findings

Connected Burau representation to moduli spaces of cone metrics

Identified kernels of specialized Burau representations in some cases

Placed the 4-strand kernel within natural topological subgroups

Abstract

We use a geometric approach to show that the reduced Burau representation specialized at roots of unity has another incarnation as the monodromy representation of a moduli space of Euclidean cone metrics on the sphere, as described by Thurston. Using the theory of orbifolds, we leverage this connection to identify the kernels of these specializations in some cases, partially addressing a conjecture of Squier. The 4-strand case is the last case where the faithfulness question for the Burau representation is unknown, a question that is related e.g. to the question of whether the Jones polynomial detects the unknot. Our results allow us to place the kernel of this representation in the intersection of several topologically natural subgroups of $B_4$.