Dehn filling, volume, and the Jones polynomial

Futer, David; Kalfagianni, Efstratia; Purcell, Jessica S.

doi:10.4310/jdg/1207834551

Cited by 101 publications

(118 citation statements)

References 31 publications

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“…One advantage of Theorem 3.11, despite its stronger hypotheses, is that the negatively curved metric can be explicitly constructed. In fact, by estimating the sectional curvatures and volume of this negatively curved metric, one may obtain explicit estimates on the hyperbolic volume of K and its Dehn fillings [8].…”

Section: Suppose That Every Twist Region Of D(k) Contains At Least 7 mentioning

confidence: 99%

Links with no exceptional surgeries

Futer¹,

Purcell²

2007

Comment. Math. Helv.

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Abstract. We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link. Mathematics Subject Classification (2000). 57M25, 57M5.

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Section: Suppose That Every Twist Region Of D(k) Contains At Least 7 mentioning

confidence: 99%

Links with no exceptional surgeries

Futer¹,

Purcell²

2007

Comment. Math. Helv.

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“…The Euclidean geometry of ∂C is an important invariant that carries a wealth of information about Dehn fillings of M . For example, if a slope s (an isotopy class of simple closed curve on ∂C) is sufficiently long, then Dehn filling M along s produces a hyperbolic manifold [3,27], whose volume can be estimated in terms of the length (s) [20].…”

Section: 2mentioning

confidence: 99%

Cusp geometry of fibered 3-manifolds

Futer¹,

Schleimer²

2014

ajm

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Abstract. Let F be a surface and suppose that ϕ : F → F is a pseudo-Anosov homeomorphism, fixing a puncture p of F . The mapping torus M = Mϕ is hyperbolic and contains a maximal cusp C about the puncture p.We show that the area (and height) of the cusp torus ∂C is equal to the stable translation distance of ϕ acting on the arc complex A(F, p), up to an explicitly bounded multiplicative error. Our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces. In particular, the proof of this theorem does not use any deep results from Teichmüller theory, Kleinian group theory, or the coarse geometry of A (F, p).A similar result holds for quasi-Fuchsian manifolds N ∼ = F × R. In that setting, we find a combinatorial estimate for the area (and height) of the cusp annulus in the convex core of N , up to explicitly bounded multiplicative and additive error. As an application, we show that covers of punctured surfaces induce quasi-isometric embeddings of arc complexes.

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“…Manipulating these polyhedra allows us to determine geometric information on the links, including bounds on volume, cusp shape and cusp area. These have been used to bound exceptional surgeries on knots [7], volumes of knots [6], and cusp shapes [13]. Moreover, the polyhedral decomposition enabled Chesebro, DeBlois, and Wilton to show that fully augmented links satisfy the virtually fibered conjecture [3].…”

Section: K Be a Link With Diagram D(k) Regard D(k)mentioning

confidence: 99%