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.
THE CONWAY POLYNOMIAL
WII SPANNING SURFACES AND SE IFERT PAIRING
RIBBONS AND SL ICES
CYCLIC BRANCHED COWERINGS
GSIGNATURE THEOREM FOR FOURMANIFOLDS
SIGNATURE OF CYCLIC BRANCHED COVERINGS