On Knots. (AM-115), Volume 115

Louis H. Kauffman

• Publisher: Princeton University Press
• Year: 2016
• ISBN:
  9781400882137
• Topic: Matematica
• Subtopic: Topologia

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.

Table of contents

• Cover
• Title
• Copyright
• Dedication
• CONTENTS
• PREFACE
• I. INTRODUCTION
• II. LINKING NUMBERS AND REIDEMEISTER MOVES
• III. THE CONWAY POLYNOMIAL
• IV. EXAMPLES AND SKEIN THEORY
• V. DETECTING SLICES AND RIBBONS, A FIRST PASS
• VI. MISCELLANY
• VII. SPANNING SURFACES AND SEIFERT PAIRING
• VIII. RIBBONS AND SLICES
• IX. ALEXANDER POLYNOMIAL AND BRANCHED COVERINGS
• X. ALEXANDER POLYNOMIAL AND ARF INVARIANT
• XI. FREE DIFFERENTIAL CALCULUS
• XII. CYCLIC BRANCHED COVERINGS
• XIII. SIGNATURE THEOREMS
• XIV. G-SIGNATURE THEOREM FOR FOUR-MANIFOLDS
• XV. SIGNATURE OF CYCLIC BRANCHED COVERINGS
• XVI. AN INVARIANT FOR COVERINGS
• XVII. SLICE KNOTS
• XVIII. CALCULATING σr FOR GENERALIZED STEVEDORE'S KNOTS
• XIX. SINGULARITIES KNOTS AND BRIESKORN VARIETIES
• APPENDIX: Generalized Polynomials and a States Model for the Jones Polynomial
• TABLES: Knot Tables and the L-Polynomial
• REFERENCES