Digraphs and cycle polynomials for free-by-cyclic groups

Let $\phi \in \mbox{Out}(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $\phi$ determines a free-by-cyclic group $\Gamma=F_n \rtimes_\phi \mathbb Z,$ and a homomorphism $\alpha \in H^1(\Gamma; \mathbb Z)$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, $\alpha$ has an open cone neighborhood $\mathcal A$ in $H^1(\Gamma;\mathbb R)$ whose integral points correspond to other fibrations of $\Gamma$ 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 $\mathcal A$.