Veering triangulations and the Thurston norm: homology to isotopy

Landry, Michael P.

doi:10.48550/arxiv.2006.16328

Cited by 3 publications

(8 citation statements)

References 14 publications

(21 reference statements)

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“…Partial branched surfaces and a result of Landry. To apply results from [Lan20] we replace M with the corresponding compact manifold as follows: First, let M be the manifold obtained by truncating the cusps of the tetrahedra of τ , as in Section 2.1. We continue to use τ p2q to denote the corresponding branched surface with boundary in M .…”

Section: Faces Of the Thurston Norm Ballmentioning

confidence: 99%

“…In the terminology of [Lan20], τ p2q is a partial branched surface of M with respect to U " M intp M q. A properly embedded surface S Ă M is carried by the partial branched surface τ p2q Ă M if S has no components completely contained in U , S intpU q Ă M is carried by τ p2q , and each component of S X U is a properly embedded π 1 -injective annulus in U with either one or both boundary components on B M .…”

Section: Faces Of the Thurston Norm Ballmentioning

confidence: 99%

“…As a last remark, we note that by [Lan20] if M is a closed manifold obtained from M by Dehn filling along slopes intersecting the ladderpoles (see Section 2.1.2) of BM enough times, the image of τ p2q under the inclusion j : M ãÑ M determines a face F of the Thurston norm ball. Thus j ˚pV τ q is an object associated to F generalizing the Teichmüller polynomial.…”

Section: Introductionmentioning

confidence: 99%

“…The veering structure of τ determines (and is determined by) the structure of these induced train tracks. We will recall some facts here that are needed for Section 5.1.1, but we refer the reader to [FG13] and [Lan20] for a more detailed analysis.…”

Section: Introductionmentioning

confidence: 99%

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A polynomial invariant for veering triangulations

Landry,

Minsky,

Taylor

2020

Preprint

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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 .

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Section: Faces Of the Thurston Norm Ballmentioning

confidence: 99%

Section: Faces Of the Thurston Norm Ballmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A polynomial invariant for veering triangulations

Landry,

Minsky,

Taylor

2020

Preprint

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“…More relevant to this paper is the work of Landry [Lan18,Lan19,Lan20] which studies the surfaces carried by the underlying 2-skeleton of the veering triangulation. This connects to our previous work [LMT20] introducing the veering polynomial, relating it to the Teichmüller polynomial, and laying the combinatorial groundwork for what is done here (although we emphasize that this paper can be read independently of the previous).…”

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confidence: 99%

Flows, growth rates, and the veering polynomial

Landry¹,

Minsky²,

Taylor³

2021

Preprint

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For certain pseudo-Anosov flows ϕ on closed 3-manifolds, unpublished work of Agol-Guéritaud produces a veering triangulation τ on the manifold M obtained by deleting ϕ's singular orbits. We show that τ can be realized in M so that its 2-skeleton is positively transverse to ϕ, and that the combinatorially defined flow graph Φ embedded in M uniformly codes ϕ'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 ϕ'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 ϕ 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 cutopen 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.

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The taut polynomial and the Alexander polynomial

Parlak

2023

Journal of Topology

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Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmüller 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. Thus, the Teichmüller polynomials are just specialisations of twisted Alexander polynomials. We also give formulae relating the taut polynomial and the untwisted Alexander polynomial. There are two formulae, depending on whether the maximal free abelian cover of a veering triangulation is edge-orientable or not. Furthermore, we consider 3-manifolds obtained by Dehn filling a veering triangulation. In this case, we give formulae that relate the specialisation of the taut polynomial under a Dehn filling and the Alexander polynomial of the Dehnfilled manifold. This extends a theorem of McMullen connecting the Teichmüller polynomial and the Alexander polynomial to the non-fibred 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.M S C 2 0 2 0 57K31 (primary), 37E30 (secondary)

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Veering triangulations and the Thurston norm: homology to isotopy

Cited by 3 publications

References 14 publications

A polynomial invariant for veering triangulations

A polynomial invariant for veering triangulations

Flows, growth rates, and the veering polynomial

The taut polynomial and the Alexander polynomial

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