The volume of hyperbolic alternating link complements

Lackenby, Marc

doi:10.1112/s0024611503014291

Cited by 136 publications

(263 citation statements)

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“…The upper bound on volume given by Lackenby, Agol and D Thurston is also linear in the number of twist regions of the diagram [10]. Specifically, Agol and Thurston showed Volume.S…”

Section: Theorem 12mentioning

confidence: 97%

“…Links with added crossing circles have been studied by many people, including Adams [1]. These links were used by Lackenby, and also by Agol and D Thurston to improve Lackenby's volume results for alternating links [10]. Provided the original diagram of K was prime and twist reduced with at least two twist regions, then the link with crossing circles added is known to be hyperbolic.…”

Section: Theorem 12mentioning

confidence: 99%

“…In order to state our results, we review some definitions concerning the diagram of a knot or link, following Lackenby [10].…”

Section: Introductionmentioning

confidence: 99%

“…In their appendix to Lackenby's paper [10], Agol and D Thurston describe how to decompose the link complement S 3 L into totally geodesic ideal polyhedra when L happens to have no crossings of strands in the projection planes (ie in our case, when K had an even number of crossings at each twist region). Their methods immediately extend to the case when single crossings of the strands in the projection plane are allowed at each twist region.…”

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confidence: 99%

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Volumes of highly twisted knots and links

Purcell

2007

Algebr. Geom. Topol.

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We show that for a large class of knots and links with complements in S 3 admitting a hyperbolic structure, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram. C is at most 113. 57M25, 57M501 IntroductionGiven a diagram of a knot or link, our goal is to determine geometric information about the complement of that link in S 3 . In particular, if the complement admits a hyperbolic structure, then by Mostow-Prasad rigidity that structure is unique. We ought to be able to make explicit statements about the geometry of this link complement, including statements about its volume. However, such results based purely on a diagram seem to be rare. For particular classes of knots and links, other results on volume have been determined. Lackenby proved that in the special case in which a knot or link is alternating, the volume of the complement is bounded above and below by the twist number of a diagram [10]. In fact, the upper bound is valid for all knots, not just alternating. This upper bound was further improved by Agol and D Thurston in an appendix to Lackenby's paper. Additionally, they found a sequence of links with volume approaching the upper bound.Recently, the lower bound has been improved by work of Agol, Storm and Thurston [2]. The proof of this result still requires that the links in question be alternating, however.

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“…The upper bound on volume given by Lackenby, Agol and D Thurston is also linear in the number of twist regions of the diagram [10]. Specifically, Agol and Thurston showed Volume.S…”

Section: Theorem 12mentioning

confidence: 97%

Section: Theorem 12mentioning

confidence: 99%

“…In order to state our results, we review some definitions concerning the diagram of a knot or link, following Lackenby [10].…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Volumes of highly twisted knots and links

Purcell

2007

Algebr. Geom. Topol.

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“…Diao et al (2005), Arsuaga et al (2007b) and Diao et al (2010) and Obeidin (2016) used this observation to construct models for prime alternating knots and links. Except for the (2, n)-torus these are hyperbolic links, whose volume can be read off the diagram up to a multiplicative constant (Lackenby 2004). Taking the uniform distribution over prime alternating link diagrams, the expected hyperbolic volume is linear in the crossing number (Obeidin 2016).…”

Section: Random Planar Curvesmentioning

confidence: 99%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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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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The volume of hyperbolic alternating link complements

Cited by 136 publications

References 6 publications

Volumes of highly twisted knots and links

Volumes of highly twisted knots and links

Models of random knots

A Survey of Hyperbolic Knot Theory

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