A polynomial invariant for veering triangulations

Abstract

We introduce a polynomial invariant Vτ P ZrH1pM q{torsions 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, Vτ 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 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 Vτ in terms of the 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 .

The taut polynomial and the Alexander polynomial

“…The connection between veering triangulations and pseudo-Anosov flows inspires a novel direction of research: studying pseudo-Anosov flows via veering triangulations. This idea is exploited by Landry, Minsky and Taylor in [10,11]. In [11] they introduced two polynomial invariants of veering triangulations -the taut polynomial and the veering polynomial.…”

“…This idea is exploited by Landry, Minsky and Taylor in [10,11]. In [11] they introduced two polynomial invariants of veering triangulations -the taut polynomial and the veering polynomial. In [10] they showed that these polynomials carry information about the exponential growth rate of the closed orbits of the associated pseudo-Anosov flows [10].…”

“…This invariant has been introduced by McMullen in [13,Section 3] and is associated to a fibred face of the Thurston norm ball. As explained in [11,Proposition 7.2] every Teichmüller polynomial can be expressed as the image of the taut polynomial of a layered veering triangulation under a fibre-preserving Dehn filling. Theorem 6.3 can therefore be interpreted as a theorem which relates the Teichmüller polynomial Θ F of a fibred face F in H 1 (N ; R) with the Alexander polynomial ∆ N of N .…”

“…(2) Theorem 6.3 does not require a veering triangulation to be layered. In particular, when a veering triangulation V of M is not layered, but does carry a surface, it gives a relation between an invariant of a non-fibred face of the Thurston norm ball in H 1 (M ; R) (see [11,Theorem 5.12]) and the Alexander polynomial of M .…”

Computation of the taut, the veering and the Teichmüller polynomials

“…This paper relies heavily on the work of Landry, Minsky and Taylor [18]. They defined the taut polynomial, the veering polynomial and the flow graph of a (transverse taut) veering triangulation.…”