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 Teichmüller polynomial of the fibered face canonically associated to \tau . Via Dehn filling, this gives a combinatorial description of the Teichmüller 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 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 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 ball determined by \tau .

The taut polynomial and the Alexander polynomial

“…It is also conjectured that any 3‐manifold admits only finitely many (possibly zero) veering triangulations. This finiteness does not follow from the result of Landry–Minsky–Taylor [25, Theorem 5.12] connecting veering triangulations to faces of the Thurston norm ball, because there are veering triangulations that determine empty faces of the Thurston norm ball (they do not carry any surfaces). Thus, compactness of the Thurston norm ball is not sufficient here.…”

“…More generally, veering triangulations have important links to the dynamics on 3‐manifolds [11, 24, 25], as well as to hyperbolic geometry [14, 15, 17, 18, 40] and the Thurston norm [21–23]. Recently, Landry, Minsky and Taylor introduced two polynomial invariants of veering triangulations, the taut polynomial and the veering polynomial [25]. The former can be used to generalise McMullen's Teichmüller polynomial [26] to the non‐fibred setting, and is the main object of study in this paper.…”

“…It plays an important role in understanding stretch factors of pseudo‐Anosov homeomorphisms of surfaces. By a result of Landry, Minsky and Taylor [25, Proposition 7.2], every Teichmüller polynomial can be expressed as a specialisation of the taut polynomial of a layered veering triangulation. In particular, Theorem 5.7 implies that Teichmüller polynomials are just specialisations of twisted Alexander polynomials and thus can be computed using Fox calculus.…”

“…Furthermore, by [25, Proposition 7.2] Theorem 6.3 can be interpreted as a theorem that relates the Teichmüller polynomial $$\Theta _{\texttt {F}}$$ of a fibred face $$\mathtt {F}$$ in $$H^1(N;\mathbb {R})$$ with the Alexander polynomial $$\Delta _N$$ of $$N$$. In [26, Theorem 7.1], McMullen proved the following.…”

“…In Corollary 7.6 we show that transverse orientability of this lamination in $$N^{fab}$$ is a sufficient condition for equality between the fibred face and a face of the Alexander norm ball. Theorem 6.3 does not require a veering triangulation to be layered. When $$\mathcal {V}$$ is not layered, but does carry a surface, it determines a (not necessarily top‐dimensional) non‐fibred face of the Thurston norm ball [25, Theorem 5.12]. In this case, we obtain a relation between the Alexander polynomial of a 3‐manifold and a polynomial invariant of a non‐fibred face of its Thurston norm ball.…”