Experimental Statistics of Veering Triangulations

Worden, William

doi:10.1080/10586458.2018.1437850

Cited by 6 publications

(5 citation statements)

References 20 publications

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“…A proof of Conjecture 4.6, even for this special family, would drastically expand the list of link complements for which we have practical, combinatorial volume estimates. See Worden [84] for more on this problem. 4.2.2.…”

Section: Lower Bounds On Volume By Results Of Jorgensen and Thurstonmentioning

confidence: 99%

A Survey of Hyperbolic Knot Theory

Futer

Kalfagianni

Purcell

2019

Knots, Low-Dimensional Topology and Applications

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We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.2010 Mathematics Subject Classification. 57M25, 57M27, 57M50.

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Section: Lower Bounds On Volume By Results Of Jorgensen and Thurstonmentioning

confidence: 99%

A Survey of Hyperbolic Knot Theory

Futer

Kalfagianni

Purcell

2019

Knots, Low-Dimensional Topology and Applications

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“…It is now clear that geometric veering triangulations are exceedingly rare. This was shown experimentally by Worden [50], who tested over 800,000 examples on a high-performance computing cluster. Given a hyperbolic surface S of complexity ξpSq ě 2, he found that for randomly sampled long words in ModpSq, the probability of the associated veering triangulation being geometric decays exponentially with the length of the word.…”

Section: Introductionmentioning

confidence: 91%

Random veering triangulations are not geometric

Futer,

Taylor,

Worden

2018

Preprint

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Every pseudo-Anosov mapping class ϕ defines an associated veering triangulation τϕ of a punctured mapping torus. We show that generically, τϕ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.

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“…This line of research led to new results concerning the exponential growth rate of closed orbits of semi-flows on sutured manifolds [24] and Birkhoff sections for pseudo-Anosov flows [38]. More generally, veering triangulations have important links to the dynamics on 3-manifolds [11,24,25], as well as to hyperbolic geometry [14,15,17,18,40] and the Thurston norm [21][22][23]. Recently, Landry, Minsky and Taylor introduced two polynomial invariants of veering triangulations, the taut polynomial and the veering polynomial [25].…”

Section: Introductionmentioning

confidence: 99%

“…More generally, veering triangulations have important links to the dynamics on 3‐manifolds [11, 24, 25], as well as to hyperbolic geometry [14, 15, 17, 18, 40] and the Thurston norm [21–23]. Recently, Landry, Minsky and Taylor introduced two polynomial invariants of veering triangulations, the taut polynomial and the veering polynomial [25].…”

Section: Introductionmentioning

confidence: 99%

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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Experimental Statistics of Veering Triangulations

Cited by 6 publications

References 20 publications

A Survey of Hyperbolic Knot Theory

A Survey of Hyperbolic Knot Theory

Random veering triangulations are not geometric

The taut polynomial and the Alexander polynomial

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