On Knots

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Princeton University Press, 1987 - Mathematics - 480 pages
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On Knots is a journey through the theory of knots, starting from the simplest combinatorial ideas--ideas arising from the representation of weaving patterns. From this beginning, topological invariants are constructed directly: first linking numbers, then the Conway polynomial and skein theory. This paves the way for later discussion of the recently discovered Jones and generalized polynomials. The central chapter, Chapter Six, is a miscellany of topics and recreations. Here the reader will find the quaternions and the belt trick, a devilish rope trick, Alhambra mosaics, Fibonacci trees, the topology of DNA, and the author's geometric interpretation of the generalized Jones Polynomial.

Then come branched covering spaces, the Alexander polynomial, signature theorems, the work of Casson and Gordon on slice knots, and a chapter on knots and algebraic singularities.The book concludes with an appendix about generalized polynomials.

 

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This is an excellent and clear book to understand knot theory. It is easy to read and easy to understand. It is great for self study and also for research in knot theory.

Contents

INTRODUCTION
3
LINKING NUMBERS AND REIDEMEISTER MOVES
9
THE CONWAY POLYNOMIAL
19
EXAMPLES AND SKEIN THEORY
42
DETECTING SLICES AND RIBBONS A FIRST PASS
70
MISCELLANY
93
Rope Trick
98
Topological Script
100
The Mobius Band
152
The Generalized Polynomial
155
The Generalized Polynomial and Regular Isotopy
163
Twisted Bands
179
SPANNING SURFACES AND SEIFERT PAIRING
181
RIBBONS AND SLICES
208
ALEXANDER POLYNOMIAL AND BRANCHED COVERINGS
229
ALEXANDER POLYNOMIAL AND ARF INVARIANT
252

Calculi
103
Infinite Forms
106
Quandles
110
Topology of DNA
113
Knots Are Decorated Fibonacci Trees
115
Alhambra Mosaic
120
Odd Knot
121
Pilars Family Tree
122
The Untwisted Double of the Double of the Figure Eight Knot
123
Applied ScriptA Ribbon Surface
124
Kirkhoffs Matrix Tree Theorem
129
States and Trails
132
The Map Theorem
147
FREE DIFFERENTIAL CALCULUS
262
CYCLIC BRANCHED COVERINGS
271
SIGNATURE THEOREMS
299
GSIGNATURE THEOREM FOR FOURMANIFOLDS
327
SIGNATURE OF CYCLIC BRANCHED COVERINGS
332
AN INVARIANT FOR COVERINGS
337
SLICE KNOTS
345
CALCULATING a FOR GENERALIZED STEVEDORES KNOTS T
355
SINGULARITIES KNOTS AND BRIESKORN VARIETIES
366
Generalized Polynomials and a States Model for the Jones Polynomial
417
Knot Tables and the LPolynomial
444
REFERENCES
474
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About the author (1987)

Kauffman-University of Illinois, Chicago

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