Characterizing right-veering homeomorphisms of the punctured torus via the Burau representation

TL;DR

This paper classifies right-veering homeomorphisms of the punctured torus using the Burau representation of the 3-strand braid group, providing a method to identify their veering behavior efficiently.

Contribution

It introduces a novel approach to classify right-veering homeomorphisms via the Burau representation, especially for reducible and periodic classes in B_3.

Findings

Reducible and periodic classes in B_3 can be identified as right-veering using the Burau representation.

Provides a quick method to determine the action of any braid in B_3 on the fundamental group.

Establishes a link between braid group representations and veering properties of homeomorphisms.

Abstract

We classify right-veering homeomorphisms of the once-punctured torus using the Burau representation of the 3-strand braid group. We show that reducible and periodic mapping classes in B_3 can be identified as right-veering by consideration of the reduced version of the Burau representation. Given any element beta in B_3, we give a method to quickly determine its action on the generators of the fundamental group of the 3-times punctured disk. This action which determines whether beta is right-veering, left-veering, or neither.