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 Teichmuller polynomial of the fibered faces canonically associated to $\tau$. Via Dehn filling, this gives a combinatorial description of the Teichmuller 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 $\textit{equal}$ 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 $\textit{flow graph}$ 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 determined by $\tau$.

pdf/a-polynomial-invariant-for-veering-triangulations-23j0p23oe4.pdf

A polynomial invariant for veering triangulations

Landry, Minsky and Taylor [LMT] introduced two polynomial invariants of veering triangulations -- the taut polynomial and the veering polynomial. We give algorithms to compute these invariants. In their definition [LMT] use only the upper track of the veering triangulation, while we consider both the upper and the lower track. We prove that the lower and the upper taut polynomials are equal. However, we show that there are veering triangulations whose lower and upper veering polynomials are different. [LMT] related the Teichmuller polynomial of a fibred face of the Thurston norm ball with the taut polynomial of the associated layered veering triangulation. We use this result to give an algorithm to compute the Teichmuller polynomial of any fibred face of the Thurston norm ball.

pdf/computation-of-the-taut-the-veering-and-the-teichm-uller-2ciggkfypd.pdf

Computation of the taut, the veering and the Teichm\"uller polynomials.

For certain pseudo-Anosov flows $\phi$ on closed $3$-manifolds, unpublished work of Agol--Gueritaud produces a veering triangulation $\tau$ on the manifold $M$ obtained by deleting $\phi$'s singular orbits. We show that $\tau$ can be realized in $M$ so that its 2-skeleton is positively transverse to $\phi$, and that the combinatorially defined flow graph $\Phi$ embedded in $M$ uniformly codes $\phi$'s orbits in a precise sense. Together with these facts we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of $\phi$'s closed orbits after cutting $M$ along certain transverse surfaces, thereby generalizing work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of $M$. 
Our work can be used to study the flow $\phi$ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the `positive' cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.

Flows, growth rates, and the veering polynomial

Generating the mapping class group of a nonorientable surface by crosscap transpositions

Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmuller polynomial of a fibred face of the Thurston norm ball. We prove that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. Then we give formulas relating the taut polynomial and the untwisted Alexander polynomial. There are two formulas; one holds when the maximal free abelian cover of a veering triangulation is edge-orientable, another holds when it is not edge-orientable. Furthermore, we consider 3-manifolds obtained by Dehn filling a veering triangulation. In this case we give a formula that relates the specialisation of the taut polynomial under the Dehn filling and the Alexander polynomial of the Dehn-filled manifold. This extends a theorem of McMullen connecting the Teichmuller polynomial and the Alexander polynomial to the nonfibred setting, and improves it in the fibred case. We also prove a sufficient and necessary condition for the existence of an orientable fibred class in the cone over a fibred face of the Thurston norm ball. 

https://onlinelibrary.wiley.com/doi/pdfdirect/10.1112/topo.12302

The taut polynomial and the Alexander polynomial

Let $N_{g}$ denote a closed nonorientable surface of genus $g$. For $g \geq 2$ the mapping class group $\mathcal{M}(N_{g})$ is generated by Dehn twists and one crosscap slide ($Y$-homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists have nontrivial roots. We give necessary and sufficient conditions for the existence of a root of a crosscap slide and a crosscap transposition.

/pdf/roots-of-crosscap-slides-and-crosscap-transpositions-3oand80qtv.pdf

Roots of crosscap slides and crosscap transpositions

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist t_c about a nonseparating circle c in the mapping class group M(N_g) of a nonorientable surface N_g of genus g. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer n>1, then for each sufficiently large g, t_c has a root of degree n in M(N_g). Moreover, for any possible degree n we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in N_g.

Roots of Dehn twists on nonorientable surfaces

Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmuller polynomial of a fibred face of the Thurston norm ball. We prove that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. Then we give formulas relating the taut polynomial and the untwisted Alexander polynomial. There are two formulas; one holds when the maximal free abelian cover of a veering triangulation is edge-orientable, another holds when it is not edge-orientable. 
Furthermore, we consider 3-manifolds obtained by Dehn filling a veering triangulation. In this case we give a formula that relates the specialisation of the taut polynomial under the Dehn filling and the Alexander polynomial of the Dehn-filled manifold. This extends a theorem of McMullen connecting the Teichmuller polynomial and the Alexander polynomial to the nonfibred setting, and improves it in the fibred case. We also prove a sufficient and necessary condition for the existence of an orientable fibred class in the cone over a fibred face of the Thurston norm ball.

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist tc about a nonseparating circle c in the mapping c...

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Papers

Computation of the taut, the veering and the Teichm\"uller polynomials.

Roots of crosscap slides and crosscap transpositions

Roots of Dehn twists on nonorientable surfaces

The taut polynomial and the Alexander polynomial

Roots of Dehn twists on nonorientable surfaces