Computation of the taut, the veering and the Teichmüller polynomials

Parlak, Anna

doi:10.48550/arxiv.2009.13558

Cited by 2 publications

(3 citation statements)

References 16 publications

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“…However, if the triangulation is veering the lower and upper tracks restricted to the faces of a single tetrahedron t are determined by the colours of the diagonal edges of t; see the lemma below. Lemma 3.8 (Lemma 3.9 in [18]). Let V = (T , α, ν) be a veering triangulation.…”

Section: The Lower and Upper Tracksmentioning

confidence: 99%

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The taut polynomial and the Alexander polynomial

Parlak¹

2021

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Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. We prove that when a veering triangulation is edge-orientable then its taut polynomial is equal to the Alexander polynomial of the underlying manifold. For triangulations which are not edge-orientable we give a sufficient condition for the equality between the support of the taut polynomial and that of the Alexander polynomial. We also consider Dehn fillings of 3-manifolds equipped with a veering triangulation. In this case we compare the image of the taut polynomial under a Dehn filling and the Alexander polynomial of the Dehnfilled manifold.As an application we extend the results of McMullen which relate the Teichmüller polynomial with the Alexander polynomial. Using this we find first cohomology classes corresponding to fibrations whose monodromy determines orientable invariant laminations in the fibre.

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Section: The Lower and Upper Tracksmentioning

confidence: 99%

“…Remark. An algorithm to compute the Teichmüller polynomial using Lemma 7.1 is given in [18,Section 8]. 7.3.…”

Section: Consequences For the Teichmüller Polynomial And Fibred Facesmentioning

confidence: 99%

The taut polynomial and the Alexander polynomial

Parlak¹

2021

Preprint

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“…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). Also, Parlak has recently introduced and implemented algorithms to compute the veering polynomial and its relatives [Par20] and demonstrated a connection with the Alexander polynomial [Par21], thereby generalizing work of McMullen on the Teichmüller polynomial [McM00].…”

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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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Computation of the taut, the veering and the Teichmüller polynomials

Cited by 2 publications

References 16 publications

The taut polynomial and the Alexander polynomial

The taut polynomial and the Alexander polynomial

Flows, growth rates, and the veering polynomial

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