arXiv subjects

Anna Parlak

Publications and source records attributed to Anna Parlak.

6 recordsLinked to original sources

arXiv2023

Arbitrarily large veering triangulations with a vanishing taut polynomial

Landry, Minsky, and Taylor introduced an invariant of veering triangulations called the taut polynomial. Via a connection between veering triangulations and pseudo-Anosov flows, it generalizes the Teichmüller polynomial of a fibered face of the Thurston norm ball to (some) non-fibered faces. We construct a sequence of veering triangulations, with the number of tetrahedra tending to infinity, whose taut polynomials vanish. These veering triangulations encode non-circular Anosov flows transverse to tori.

math.GT↗

arXiv2023

The taut polynomial and the Alexander polynomial

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.

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arXiv2023

Mutations and faces of the Thurston norm ball dynamically represented by multiple distinct flows

A pseudo-Anosov flow on a hyperbolic 3-manifold dynamically represents a face F of the Thurston norm ball if the cone on F is dual to the cone spanned by homology classes of closed orbits of the flow. Fried showed that for every fibered face of the Thurston norm ball there is a unique, up to isotopy and reparametrization, flow which dynamically represents the face. Using veering triangulations we have found that there are non-fibered faces of the Thurston norm ball which are dynamically represented by multiple topologically inequivalent flows. This raises a question of how distinct flows representing the same face are related. We define combinatorial mutations of veering triangulations along surfaces that they carry. We give sufficient and necessary conditions for the mutant triangulation to be veering. After appropriate Dehn filling these veering mutations correspond to transforming one 3-manifold M with a pseudo-Anosov flow transverse to an embedded surface S into another 3-manifold admitting a pseudo-Anosov flow transverse to a surface homeomorphic to S. We show that a non-fibered face of the Thurston norm ball can be dynamically represented by two distinct flows that differ by a veering mutation.

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arXiv2021

Computation of the taut, the veering and the Teichmüller polynomials

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 Teichmüller 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 Teichmüller polynomial of any fibred face of the Thurston norm ball.

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arXiv2017

Roots of Dehn twists on nonorientable surfaces

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.

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arXiv2017

Roots of crosscap slides and crosscap transpositions

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.

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