Digraphs and cycle polynomials for free-by-cyclic groups

Let $\varphi \in Out(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $\varphi$ determines a free-by-cyclic group $\mathrm{\Gamma }=F_n{⋊}_{\varphi }\mathbb{Z},$ and a homomorphism $\alpha \in H^1(\mathrm{\Gamma };\mathbb{Z})$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, $\alpha$ has an open cone neighborhood $\mathcal{A}$ in $H^1(\mathrm{\Gamma };\mathbb{R})$ whose integral points correspond to other fibrations of $\mathrm{\Gamma }$ whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen’s Teichm"uller polynomial that computes the dilatations of all outer automorphism in $\mathcal{A}$.