Polynomial invariants of pseudo-Anosov maps

TL;DR

This paper introduces three new polynomial invariants derived from the characteristic polynomial of transition matrices associated with pseudo-Anosov maps, aiding in distinguishing maps with identical dilatations.

Contribution

It presents three novel polynomial invariants of pseudo-Anosov maps, with explicit formulas for their degrees and methods for computation from train tracks.

Findings

Three new polynomial invariants are introduced.

Invariants help distinguish pseudo-Anosov maps with same dilatation.

Formulas for degrees of invariants are provided.

Abstract

We investigate the structure of the characteristic polynomial det(xI-T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial invariants of [F], one of them being the product of the other two, and all three being divisors of det(xI-T). The degrees of the new polynomials are invariants of [F ] and we give simple formulas for computing them by a counting argument from an invariant train track. We give examples of genus 2 pseudo-Anosov maps having the same dilatation, and use our invariants to distinguish them.