Hyperbolic Knot Theory (Graduate Studies in Mathematics)

This book provides an introduction to hyperbolic geometry in dimension three, with motivation and applications arising from knot theory. Hyperbolic geometry was first used as a tool to study knots by Riley and then Thurston in the 1970s. By the 1980s, combining work of Mostow and Prasad with Gordon and Luecke, it was known that a hyperbolic structure on a knot complement in the 3-sphere gives a complete knot invariant. However, it remains a difficult problem to relate the hyperbolic geometry of a knot to other invariants arising from knot theory. In particular, it is difficult to determine hyperbolic geometric information from a knot diagram, which is classically used to describe a knot. This textbook provides background on these problems, and tools to determine hyperbolic information on knots. It also includes results and state-of-the art techniques on hyperbolic geometry and knot theory to date. The book was written to be interactive, with many examples and exercises. Some important results are left to guided exercises. The level is appropriate for graduate students with a basic background in algebraic topology, particularly fundamental groups and covering spaces. Some experience with some differential topology and Riemannian geometry will also be helpful. Contents Preface Why I wrote this book How I structured the book Prerequisites and notes to students Acknowledgments Introduction Chapter 0. A Brief Introduction to Hyperbolic Knots 0.1. An introduction to knot theory 0.2. Problems in knot theory 0.3. Exercises Part 1 . Foundations of Hyperbolic Structures Chapter 1. Decomposition of the Figure-8 Knot 1.1. Polyhedra 1.2. Generalizing: Exercises Chapter 2. Calculating in Hyperbolic Space 2.1. Hyperbolic geometry in dimension two 2.2. Hyperbolic geometry in dimension three 2.3. Exercises Chapter 3. Geometric Structures on Manifolds 3.1. Geometric structures 3.2. Complete structures 3.3. Developing map and completeness 3.4. Exercises Chapter 4. Hyperbolic Structures and Triangulations 4.1. Geometric triangulations 4.2. Edge gluing equations 4.3. Completeness equations 4.4. Computing hyperbolic structures 4.5. Exercises Chapter 5. Discrete Groups and the Thick-Thin Decomposition 5.1. Discrete subgroups of hyperbolic isometries 5.2. Elementary groups 5.3. Thick and thin parts 5.4. Hyperbolic manifolds with finite volume 5.5. Universal elementary neighborhoods 5.6. Exercises Chapter 6. Completion and Dehn Filling 6.1. Mostow–Prasad rigidity 6.2. Completion of incomplete structures 6.3. Hyperbolic Dehn filling space 6.4. A brief summary of geometric convergence 6.5. Exercises 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 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