دانلود کتاب Hyperbolic Knot Theory

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Contents 8 Preface 12 Why I wrote this book 12 How I structured the book 13 Prerequisites and notes to students 15 Acknowledgments 15 Introduction 18 Chapter 0. A Brief Introduction to Hyperbolic Knots 20 0.1. An introduction to knot theory 20 0.2. Problems in knot theory 23 0.3. Exercises 35 Part 1 . Foundations of Hyperbolic Structures 36 Chapter 1. Decomposition of the Figure-8 Knot 38 1.1. Polyhedra 38 1.2. Generalizing: Exercises 45 Chapter 2. Calculating in Hyperbolic Space 48 2.1. Hyperbolic geometry in dimension two 48 2.2. Hyperbolic geometry in dimension three 57 2.3. Exercises 59 Chapter 3. Geometric Structures on Manifolds 64 3.1. Geometric structures 64 3.2. Complete structures 71 3.3. Developing map and completeness 82 3.4. Exercises 83 Chapter 4. Hyperbolic Structures and Triangulations 86 4.1. Geometric triangulations 86 4.2. Edge gluing equations 90 4.3. Completeness equations 96 4.4. Computing hyperbolic structures 100 4.5. Exercises 101 Chapter 5. Discrete Groups and the Thick-Thin Decomposition 104 5.1. Discrete subgroups of hyperbolic isometries 104 5.2. Elementary groups 110 5.3. Thick and thin parts 113 5.4. Hyperbolic manifolds with finite volume 116 5.5. Universal elementary neighborhoods 118 5.6. Exercises 125 Chapter 6. Completion and Dehn Filling 128 6.1. Mostow–Prasad rigidity 128 6.2. Completion of incomplete structures 129 6.3. Hyperbolic Dehn filling space 133 6.4. A brief summary of geometric convergence 141 6.5. Exercises 147 Part 2 . Tools, Techniques, and Families of Examples 150 Chapter 7. Twist Knots and Augmented Links 152 7.1. Twist knots and Dehn fillings 152 7.2. Double twist knots and the Borromean rings 157 7.3. Augmenting and highly twisted knots 160 7.4. Cusps of fully augmented links 166 7.5. Exercises 172 Chapter 8. Essential Surfaces 176 8.1. Incompressible surfaces 176 8.2. Torus decomposition, Seifert fibering, and geometrization 182 8.3. Normal surfaces, angled polyhedra, and hyperbolicity 184 8.4. Pleated surfaces and a 6-theorem 192 8.5. Exercises 200 Chapter 9. Volume and Angle Structures 202 9.1. Hyperbolic volume of ideal tetrahedra 202 9.2. Angle structures and the volume functional 212 9.3. Leading-trailing deformations 214 9.4. The Schläfli formula 220 9.5. Consequences 221 9.6. Exercises 223 Chapter 10. Two-Bridge Knots and Links 226 10.1. Rational tangles and 2-bridge links 226 10.2. Triangulations of 2-bridge links 230 10.3. Positively oriented tetrahedra 239 10.4. Maximum in interior 246 10.5. Exercises 255 Chapter 11. Alternating Knots and Links 258 11.1. Alternating diagrams and hyperbolicity 259 11.2. Checkerboard surfaces 272 11.3. Exercises 276 Chapter 12. The Geometry of Embedded Surfaces 278 12.1. Belted sums and mutations 279 12.2. Fuchsian, quasifuchsian, and accidental surfaces 283 12.3. Fibers and semifibers 289 12.4. Exercises 295 Part 3 . Hyperbolic Knot Invariants 298 Chapter 13. Estimating Volume 300 13.1. Summary of bounds encountered so far 300 13.2. Negatively curved metrics and Dehn filling 304 13.3. Volume, guts, and essential surfaces 318 13.4. Exercises 327 Chapter 14. Ford Domains and Canonical Polyhedra 330 14.1. Horoballs and isometric spheres 331 14.2. Ford domain 338 14.3. Canonical polyhedra 345 14.4. Exercises 350 Chapter 15. Algebraic Sets and the ?-Polynomial 352 15.1. The gluing variety 352 15.2. Representations of knots 358 15.3. The ?-polynomial 366 15.4. Exercises 370 Bibliography 372 Index 382