The spectra of volume and determinant densities of links

Burton, Stephan D.

doi:10.1016/j.topol.2016.08.022

Cited by 5 publications

(6 citation statements)

References 17 publications

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“…As a corollary to the proof, we have that volume densities for links in the thickened torus T × (0, 1) are dense in the interval [0, v oct ]. This extends previous results that volume densities are dense in [0, v oct ] for knots in S 3 from [Bur15] and [ACJ + 17].…”

Section: Introductionsupporting

confidence: 89%

“…As the proof of Theorem 4.7 involves increases in genus, it cannot be employed to show density of volume densities in [0, β g ]. However, augmentation and taking a belted sum with links in S 3 leaves genus intact, and thus the set of volume densities of links in S g × I can be shown to be dense over at least the interval [0, v oct ] by employing results of, e.g., [Bur15] and [ACJ + 17].…”

Section: Solidmentioning

confidence: 99%

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Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links

Adams

Calderon

Mayer

2020

Topology and its Applications

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We present explicit geometric decompositions of the hyperbolic complements of alternating kuniform tiling links, which are alternating links whose projection graphs are k-uniform tilings of S 2 , E 2 , or H 2 . A consequence of this decomposition is that the volumes of spherical alternating k-uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral realization of the k-uniform tiling on H 2 . In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces Sg × I with genus g ≥ 2 and totally geodesic boundaries. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for all hyperbolic links in Sg × I, ranging over all g ≥ 2, is a dense subset of the interval [0, 2voct], where voct ≈ 3.66386 is the volume of the ideal regular octahedron.

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Section: Introductionsupporting

confidence: 89%

Section: Solidmentioning

confidence: 99%

Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links

Adams

Calderon

Mayer

2020

Topology and its Applications

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“…Ruberman discovered a similar result for 4-punctured spheres and related surfaces [Rub87]. Both techniques are still frequently used to build examples of hyperbolic knots and links with particular geometric properties (for example volume: [Bur16], [AKC + 17], short geodesics: [Mil17], cusp shapes: [DP19]).…”

Section: Chapter 12mentioning

confidence: 95%

Hyperbolic Knot Theory

Purcell

2020

Preprint

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Part 2. Tools, Techniques, and Families of Examples Chapter 7. Twist Knots and Augmented Links 7.1. Twist knots and Dehn fillings 7.2. Double twist knots and the Borromean rings 7.3. Augmenting and highly twisted knots 7.4. Cusps of fully augmented links 7.5. Exercises Chapter 8. Essential Surfaces 8.1. Incompressible surfaces 8.2. Torus decomposition, Seifert fibering, and geometrization 8.3. Normal surfaces, angled polyhedra, and hyperbolicity 8.4. Pleated surfaces and a 6-theorem 8.5. Exercises Chapter 9. Volume and Angle Structures 9.1. Hyperbolic volume of ideal tetrahedra 9.2. Angle structures and the volume functional 9.3. Leading-trailing deformations 9.4. The Schläfli formula 9.5. Consequences 9.6. Exercises Chapter 10. Two-Bridge Knots and Links 10.1. Rational tangles and 2-bridge links 10.2. Triangulations of 2-bridge links 10.3. Positively oriented tetrahedra 10.4. Maximum in interior 10.5. Exercises Chapter 11. Alternating Knots and Links 11.1. Alternating diagrams and hyperbolicity 11.2. Checkerboard surfaces CONTENTS v 11.3. Exercises Chapter 12. The Geometry of Embedded Surfaces 12.1. Belted sums and mutations 12.2. Fuchsian, quasifuchsian, and accidental surfaces 12.3. Fibers and semifibers 12.4. Exercises Part 3. Hyperbolic Knot Invariants Chapter 13. Estimating Volume 13.1. Summary of bounds encountered so far 13.2. Negatively curved metrics and Dehn filling 13.3. Volume, guts, and essential surfaces 13.4. Exercises Chapter 14. Ford Domains and Canonical Polyhedra 14.1. Horoballs and isometric spheres 14.2. Ford domain 14.3. Canonical polyhedra 14.4. Exercises Chapter 15. Algebraic Sets and the A-Polynomial 15.1. The gluing variety 15.2. Representations of knots 15.3. The A-polynomial 15.4. Exercises Bibliography Index Hyperbolic knots, however, are not well-understood in general, and yet they are extremely common. For example, of all prime knots up to 16 crossings, classified by Hoste, Thistlethwaite, and Weeks, 13 are torus knots, 20 are satellite knots, and the remaining 1,701,903 are hyperbolic [HTW98]. Of all prime knots up to 19 crossings, 15 are torus knots, 380 are satellite knots, and the remaining 352,151,858 are hyperbolic [Bur20]. Moreover, if a knot complement admits a hyperbolic structure, then that structure is unique, by work of Mostow and Prasad in the 1970s [Mos73, Pra73]. More carefully, Mostow showed that if there is an isomorphism between the fundamental groups of two closed hyperbolic 3-manifolds, then there is an isometry taking one to the other. Prasad extended this work to vii viii Preface Why I wrote this bookThis book is an introduction to hyperbolic geometry in three dimensions, with motivations and examples coming from the field of knots. It is also an introduction to knot theory, with tools, techniques, and topics coming from geometry. As I write, I believe it is the only book that attempts to be both.To be clear, there are dozens of excellent books on knot theory, available from undergraduate to graduate levels, many of them classics that I learned from and contin...

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“…Thus, c(W (p, q)) = q(p − 1). For example, W (5,4) is the closure of the 5-braid in Figure 3. Theorems 1.4 and 1.5 imply that any sequence of knots W (p, q), with p, q → ∞, is both geometrically and diagrammatically maximal.…”

Section: Introductionmentioning

confidence: 99%

Geometrically and diagrammatically maximal knots

Champanerkar¹,

Kofman²,

Purcell³

2016

J. London Math. Soc.

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The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios.

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The spectra of volume and determinant densities of links

Cited by 5 publications

References 17 publications

Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links

Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links

Hyperbolic Knot Theory

Geometrically and diagrammatically maximal knots

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