The knot book : an elementary introduction to the mathematical theory of knots

[First edition, reprinted with corrections.] Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. "The Knot Book" is an introduction to this rich theory, starting with our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. "The Knot Book" is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. This is a compelling book that will comfortably escort you into the marvelous world of knot theory. Whether you are a mathematics student, someone working in a related field, or an amateur mathematician, you will find much of interest in "The Knot Book". Colin Adams received the Mathematical Association of America (MAA) Award for Distinguished Teaching and has been an MAA Polya Lecturer and a Sigma Xi Distinguished Lecturer. Other key books of interest available from the "AMS" are "Knots and Links" and "The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes". Over a century old, knot theory is today one of the most active areas of modern mathematics. The study of knots has led to important applications in DNA research and the synthesis of new molecules. And it is having a significant impact on statistical mechanics and quantum field theory. Many of the problems discussed in knot theory, including those treated here, can be understood with only a background of high school algebra and can be solved by the curious amateur. All you need to begin is a piece of string, a little math, a little imagination, and Colin Adams's The Knot Book - the first book to make cutting-edge research in knot theory accessible to a nonspecialist audience. What are the different properties and classifications of knots? How do you determine whether a knot is actually knotted or can be untangled? What is the appropriate measure of the complexity of a knot? What does knot theory research offer to other sciences? In The Knot Book Colin Adams describes and illustrates the work being done to answer these questions. Starting with the simplest knot (the trivial knot or unknot), Adams guides readers through increasingly more intricate twists and turns of knot theory, exploring problems and theorems mathematicians now can solve, as well as those that remain open. He also looks at how knot theory is providing important insights in biology, chemistry, physics, and other fields. Included are hundreds of illustrations of knots (including a table at the end of the book displaying nearly 200 different knots) as well as worked examples, exercises open problems - even a few knot jokes and pastimes. Colin Adams explains knot theory with an enthusiasm and an informal style that makes this seemingly mysterious subject easy to approach. With The Knot Book and a mathematical background that includes no more than a familiarity with polynomials, you will be able to understand and work with some of the discipline's most modern and provocative ideas. Title 2 Copyright 5 Contents 8 Preface 12 Chapter 1 Introduction 16 1.1 Introduction 16 1.2 Composition of Knots 22 1.3 Reidemeister Moves 27 1.4 Links 31 1.5 Tricolorability 37 1.6 Knots and Sticks 42 Chapter 2 Tabulating Knots 46 2.1 History of Knot Tabulation 46 2.2 The Dowker Notation for Knots 50 2.3 Conway's Notation 56 2.4 Knots and Planar Graphs 66 Chapter 3 Invariants of Knots 72 3.1 Unknotting Number 72 3.2 Bridge Number 79 3.3 Crossing Number 82 Chapter 4 Surfaces and Knots 86 4.1 Surfaces without Boundary 86 4.2 Surfaces with Boundary 102 4.3 Genus and Seifert Surfaces 110 Chapter 5 Types of Knots 122 5.1 Torus Knots 122 5.2 Satellite Knots 130 5.3 Hyperbolic Knots 134 5.4 Braids 142 5.5 Almost Alternating Knots 154 Chapter 6 Polynomials 162 6.1 The Bracket Polynomial and the Jones Polynomial 162 6.2 Polynomials of Alternating Knots 170 6.3 The Alexander and HOMFLY Polynomials 180 6.4 Amphicheirality 190 Chapter 7 Biology, Chemistry and Physics 196 7.1 DNA 196 7.2 Synthesis of Knotted Molecules 210 7.3 Chirality of Molecules 216 7.4 Statistical Mechanics and Knots 220 Chapter 8 Knots, Links, and Graphs 230 8.1 Links in Graphs 230 8.2 Knots in Graphs 237 8.3 Polynomials of Graphs 246 Chapter 9 Topology 258 9.1 Knot Complements and Three-Manifolds 258 9.2 The Three-Sphere and Lens Spaces 261 9.3 The Poincaré Conjecture, Dehn Surgery, and the Gordon–Luecke Theorem 272 Chapter 10 Higher Dimensional Knotting 280 10.1 Picturing Four Dimensions 280 10.2 Knotted Spheres in Four Dimensions 287 10.3 Knotted Three-Spheres in Five-Space 288 Knot Jokes and Pastimes 291 Appendix: Table of Knots, Links, and Knot and Link Invariants 294 Suggested Readings and References 306 Index 318 Corrections to the 2004 AMS Printing 322 "Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. The Knot Book is an introduction to this rich theory, starting from our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. The Knot Book is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. This is a compelling book that will comfortably escort you into the marvelous world of knot theory. Whether you are a mathematics student, someone working in a related field, or an amateur mathematician, you will find much of interest in The Knot Book." -- Provided by publisher

Knots are familiar objects. We use them to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. The Knot Book is an introduction to this rich theory, starting from our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. The Knot Book is also about the excitement of doing mathematics. Colin Adams engages the reader with fascinating examples, superb figures, and thought-provoking ideas. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. This is a compelling book that will comfortably escort you into the marvelous world of knot theory. Whether you are a mathematics student, someone working in a related field, or an amateur mathematician, you will find much of interest in The Knot Book.