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Abstract For a pseudo-Anosov flow $$\varphi $$ without perfect fits on a closed $$3$$ -manifold, Agol–Guéritaud produce a veering triangulation $$\tau $$ on the manifold M obtained by deleting the singular orbits of $$\varphi $$ . We show that $$\tau $$ can be realized in M so that its 2-skeleton is positively transverse to $$\varphi $$ , and that the combinatorially defined flow graph $$\Phi $$ embedded in M uniformly codes the orbits of $$\varphi $$ in a precise sense. Together with these facts, we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of the closed orbits of $$\varphi $$ after cutting M along certain transverse surfaces, thereby generalizing the work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M . Our work can be used to study the flow $$\varphi $$ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the ‘positive’ cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $$3$$ -manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.