A polynomial invariant for veering triangulations

Abstract

We introduce a polynomial invariant $V_\tau \in \mathbb{Z}[H_1(M)/\text{torsion}]$ associated to a veering triangulation $\tau$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration, $V_\tau$ recovers the Teichm\"uller polynomial of the fibered faces canonically associated to $\tau$. Via Dehn filling, this gives a combinatorial description of the Teichm\"uller polynomial for any hyperbolic fibered $3$-manifold. For a general veering triangulation $\tau$, we show that the surfaces carried by $\tau$ determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is $\textit{equal}$ to the cone over a (generally non-fibered) face of the Thurston norm ball, and that $\tau$ computes the norm on this cone in a precise sense. We also give a combinatorial description of $V_\tau$ in terms of the $\textit{flow graph}$ for $\tau$ and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm determined by $\tau$.