A Simple Computation of Teichmüller Polynomials from Integer Permutations

Abstract

We present a simple method to compute the Teichmüller polynomial of the fibered face of a hyperbolic $3$-manifold $M_{\varphi }$ obtained as the mapping torus of a pseudo-Anosov homeomorphism $\varphi$ of a closed surface. We assume $\varphi$ has orientable invariant foliations and fixes each singular trajectory. We use a characterisation of such homeomorphisms in terms of a permutation of a finite set of integers to give a direct implementation of McMullens algorithm using train tracks. Train tracks with a single vertex suffice in this case. As an application, for each $p\in {\mathbb{Z}}_{\ge 0}$, we find an infinite sequence of Teichmüller polynomials ${\Theta }_{g,p}$ associated to pseudo-Anosov maps on surfaces of genus $g\ge 2$, such that the hyperbolic 3-manifold obtained as the mapping torus has first Betti number $g$. These polynomials realize a positive proportion of bi-Perron units of each degree as pseudo-Anosov stretch-factors.