Cusp shape and tunnel number

Dang, Vinh Xuan; Purcell, Jessica S.

doi:10.1090/proc/14336

Cited by 2 publications

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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: 94%

Hyperbolic Knot Theory

Purcell

2020

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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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