Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell

In \cite{Ka14} we produced an algorithm for deciding whether or not an element $\phi\in Out(F_N)$ is an iwip ("fully irreducible") automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this paper we refine the algorithm from \cite{Ka14} by eliminating these inefficient features, and also by eliminating any use of mapping class groups algorithms. Our main result is to produce, for any fixed $N\ge 3$, an algorithm which, given a topological representative $f$ of an element $\phi$ of $Out(F_N)$, decides in polynomial time in terms of the "size" of $f$, whether or not $\phi$ is fully irreducible. In addition, we provide a train track criterion of being fully irreducible which covers all fully irreducible elements of $Out(F_N)$, including both atoroidal and non-atoroidal ones. We also give an algorithm, alternative to that of Turner, for finding all the indivisible Nielsen paths in an expanding train track map, and estimate the complexity of this algorithm. An appendix by Mark Bell provides a polynomial upper bound, in terms of the size of the topological representative, on the complexity of the Bestvina-Handel algorithm \cite{BH92} for finding either an irreducible train track representative or a topological reduction.