Veering triangulations and the Thurston norm: Homology to isotopy

Abstract

We show that a veering triangulation τ specifies a face σ of the Thurston norm ball of a closed 3-manifold, and computes the Thurston norm in the cone over σ. Further, we show that τ collates exactly the taut surfaces representing classes in the cone over σ up to isotopy. The analysis includes nonlayered veering triangulations and nonfibered faces. We also prove an analogous theorem for manifolds with boundary that is integral to a theorem of Landry-Minsky-Taylor relating the Thurston norm to the veering polynomial, a new generalization of McMullen's Teichmüller polynomial.