A polynomial invariant for veering triangulations

We introduce a polynomial invariant $V_τ\in \mathbb{Z}[H_1(M)/\text{torsion}]$ associated to a veering triangulation $τ$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration, $V_τ$ recovers the Teichmüller polynomial of the fibered faces canonically associated to $τ$. Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered $3$-manifold. For a general veering triangulation $τ$, we show that the surfaces carried by $τ$ 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 $τ$ computes the norm on this cone in a precise sense. We also give a combinatorial description of $V_τ$ in terms of the $\textit{flow graph}$ for $τ$ 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 $τ$.

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