A polynomial invariant for veering triangulations
Michael P. Landry
Washington University in Saint Louis, USAYair N. Minsky
Yale University, New Haven, USASamuel J. Taylor
Temple University, Philadelphia, USA
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Abstract
We introduce a polynomial invariant associated to a veering triangulation of a -manifold . In the special case where the triangulation is layered, i.e. comes from a fibration, recovers the Teichmüller polynomial of the fibered face canonically associated to . Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered -manifold.
For a general veering triangulation , we show that the surfaces carried by determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is equal to the cone over a (generally non-fibered) face of the Thurston norm ball, and that computes the norm on this cone in a precise sense. We also give a combinatorial description of in terms of the flow graph for 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 ball determined by .
Cite this article
Michael P. Landry, Yair N. Minsky, Samuel J. Taylor, A polynomial invariant for veering triangulations. J. Eur. Math. Soc. 26 (2024), no. 2, pp. 731–788
DOI 10.4171/JEMS/1368