The Knot Book: An Elementary Introduction To The Mathematical Theory Of Knots

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Format: Paperback
Pub. Date: 2004-08-01
Publisher(s): Amer Mathematical Society
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Summary

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 withfascinating 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 mathematicsstudent, 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.

Table of Contents

Preface xi
Introduction
1(30)
Introduction
1(6)
Composition of Knots
7(5)
Reidemeister Moves
12(4)
Links
16(6)
Tricolorability
22(5)
Knots and Sticks
27(4)
Tabulating Knots
31(26)
History of Knot Tabulation
31(4)
The Dowker Notation for Knots
35(6)
Conway's Notation
41(10)
Knots and Planar Graphs
51(6)
Invariants of Knots
57(14)
Unknotting Number
57(7)
Bridge Number
64(3)
Crossing Number
67(4)
Surfaces and Knots
71(36)
Surfaces without Boundary
71(16)
Surfaces with Boundary
87(8)
Genus and Seifert Surfaces
95(12)
Types of Knots
107(40)
Torus Knots
107(8)
Satellite Knots
115(4)
Hyperbolic Knots
119(8)
Braids
127(12)
Almost Alternating Knots
139(8)
Polynomials
147(34)
The Bracket Polynomial and the Jones Polynomial
147(9)
Polynomials of Alternating Knots
156(9)
The Alexander and Homfly Polynomials
165(11)
Amphicheirality
176(5)
Biology, Chemistry, and Physics
181(34)
DNA
181(14)
Synthesis of Knotted Molecules
195(6)
Chirality of Molecules
201(4)
Statistical Mechanics and Knots
205(10)
Knots, Links, and Graphs
215(28)
Links in Graphs
215(7)
Knots in Graphs
222(9)
Polynomials of Graphs
231(12)
Topology
243(22)
Knot Complements and Three-Manifolds
243(3)
The Three-Sphere and Lens Spaces
246(11)
The Poincare Conjecture, Dehn Surgery, and the Gordon--Luecke Theorem
257(8)
Higher Dimensional Knotting
265(11)
Picturing Four Dimensions
265(7)
Knotted Spheres in Four Dimensions
272(1)
Knotted Three-Spheres in Five-Space
273(3)
Knot Jokes and Pastimes 276(3)
Appendix: Table of Knots, Links, and Knot and Link Invariants 279(12)
Suggested Readings and References 291(12)
Index 303(4)
Corrections to the 2004 AMS Printing 307

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