On Knots

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Princeton University Press, 1987 - Mathematics - 480 pages

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

THE CONWAY POLYNOMIAL
19
MISCELLANY
93
Calculi 1
104
WII SPANNING SURFACES AND SE IFERT PAIRING
181
RIBBONS AND SL ICES
229
CYCLIC BRANCHED COWERINGS
271
SIGNATURE THEOREMS
299
GSIGNATURE THEOREM FOR FOURMANIFOLDS
327
AN INVARIANT FOR COVERINGS
337
SLICE KNOTS
345
CALCULATING O FOR GENERALIZED STEVEDORES KNOTS
355
SINGULARITIES KNOTS AND BRIESKORN WARIETIES
366
Generalized Polynomials and a States Model for the Jones Polynomial 4 17
417
Knot Tables and the LPolynomial
444
REFERENCES
474
Copyright

SIGNATURE OF CYCLIC BRANCHED COVERINGS
332

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Page 476 - VFR Jones. A polynomial invariant for knots and links via Von Neumann Algebras, BAMS 12(1985), 103111 . [J02] VFR Jones.

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