A Simple Computation of Teichm\"uller Polynomials from Integer Permutations

We present a simple method to compute the Teichm\"uller polynomial of the fibered face of a hyperbolic $3$-manifold $M_\phi$ obtained as the mapping torus of a pseudo-Anosov homeomorphism $\phi$ of a closed surface. We assume $\phi$ 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 $p\in\mathbb{Z}_{\geq0}$, we find an infinite sequence of Teichm\"uller polynomials $\Theta_{g,p}$ associated to pseudo-Anosov maps on surfaces of genus $g\geq2$, 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.