Polynomial invariants of Pseudo-Anosov maps

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. © 2012 World Scientific Publishing Company.