The taut polynomial and the Alexander polynomial

Parlak, Anna

doi:10.48550/arxiv.2101.12162

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…In the orientable case, McMullen proved these invariants are closely related in that the Alexander polynomial divides the Teichmüller polynomial [McM00,Theorem 7.1]. More recently, Parlak [Par21] extended McMullen's work to explicitly determine the quotient polynomial, giving a precise relation between the two polynomials and the associated stretch factors.…”

Section: Introductionmentioning

confidence: 99%

Orientable maps and polynomial invariants of free-by-cyclic groups

Dowdall¹,

Gupta²,

Taylor³

2022

Preprint

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We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism ϕ and use it to characterize when the homological stretch factor of ϕ is equal to its geometric stretch factor. Contents 1. Introduction 1 2. Orientable graph maps and stretch factors 4 3. Orientability for free group endomorphisms 8 4. Computing the Alexander polynomial 11 5. The folded mapping torus, cross sections, and the McMullen polynomial 20 6. Relating the polynomials and applications 26 7. Examples 31 References 34

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Section: Introductionmentioning

confidence: 99%

Orientable maps and polynomial invariants of free-by-cyclic groups

Dowdall¹,

Gupta²,

Taylor³

2022

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 [Par21a], and demonstrated a connection with the Alexander polynomial [Par21b], thereby generalizing the work of McMullen on the Teichmüller polynomial [McM00].…”

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confidence: 99%

Flows, growth rates, and the veering polynomial

Landry

Minsky²,

Taylor³

2022

Ergod. Th. Dynam. Sys.

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For a pseudo-Anosov flow $\varphi $ without perfect fits on a closed $3$ -manifold, Agol–Guéritaud produce a veering triangulation $\tau $ on the manifold M obtained by deleting the singular orbits of $\varphi $ . We show that $\tau $ can be realized in M so that its 2-skeleton is positively transverse to $\varphi $ , and that the combinatorially defined flow graph $\Phi $ embedded in M uniformly codes the orbits of $\varphi $ 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 the closed orbits of $\varphi $ after cutting M along certain transverse surfaces, thereby generalizing the 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 $\varphi $ 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.

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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].…”

mentioning

confidence: 99%

Flows, growth rates, and the veering polynomial

Landry¹,

Minsky²,

Taylor³

2021

Preprint

View full text Add to dashboard Cite

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

Cited by 3 publications

References 11 publications

Orientable maps and polynomial invariants of free-by-cyclic groups

Orientable maps and polynomial invariants of free-by-cyclic groups

Flows, growth rates, and the veering polynomial

Flows, growth rates, and the veering polynomial

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