## Knots and Physics, Third EditionThis invaluable book is an introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. |

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### Contents

A Short Course of Knots and Physics | 3 |

Physical Knots | 4 |

Diagrams and Moves | 8 |

States and the Bracket Polynomial | 25 |

Alternating Links and Checkerboard Surfaces | 39 |

The Jones Polynomial and its Generalizations | 49 |

An Oriented State Model for VKt | 74 |

Braids and the Jones Polynomial | 85 |

On a Crossing | 332 |

Slide Equivalence | 336 |

Unoriented Diagrams and Linking Numbers | 339 |

The Penrose Chromatic Recursion | 346 |

The Chromatic Polynomial | 353 |

The Potts Model and the Dichromatic Polynomial | 364 |

Quaternions Cayley Numbers and the Belt Trick | 403 |

The Quaternion Demonstrator | 427 |

Abstract Tensors and the YangBaxter Equation | 104 |

Formal Feynman Diagrams Bracket as a VacuumVacuum Expectation and the Quantum Group SL2q | 117 |

The Form of the Universal 2matrix | 148 |

YangBaxter Models for Specializations of the Homfly Polynomial | 161 |

The Alexander Polynomial | 174 |

KnotCrystals Classical Knot Theory in a Modern Guise | 186 |

The Kauffman Polynomial | 215 |

Oriented Models and Piecewise Linear Models | 235 |

Three Manifold Invariants from the Jones Polynomial | 250 |

Integral Heuristics and Wittens Invariants | 285 |

Appendix Solutions to the YangBaxter Equation | 316 |

Knots and Physics Miscellany 1 Theory of Hitches | 323 |

The Rubber Band and Twisted Tube | 329 |

The Penrose Theory of Spin Networks | 443 |

QSpin Networks and the Magic Weave | 459 |

Knots and Strings Knotted Strings | 475 |

DNA and Quantum Field Theory | 488 |

Knots in Dynamical Systems The Lorenz Attractor | 501 |

Introduction | 541 |

Gauss Codes Quantum Groups and Ribbon Hopf Algebras | 551 |

Spin Networks Topology and Discrete Physics | 597 |

Link Polynomials and a Graphical Calculus | 638 |

Knots Tangles and Electrical Networks | 684 |

Knot Theory and Functional Integration | 724 |

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### Common terms and phrases

3-manifolds 4-valent Alexander polynomial ambient isotopy arcs associated bracket polynomial braid group calculation colors completes the proof components compute construction corresponding crossing defined diagrammatic dimensional edge element embedding equivalent evaluation example field theory Figure follows formalism formula functional integral given graph G Hence Homfly polynomial Hopf algebra identity index set indices isotopy invariant Jones polynomial Jordan curve knot or link knot theory knots and links L. H. Kauffman labelled Lemma Lie algebra link diagram link invariants linking number Math matrix multiplication nodes Note obtained oriented link Phys planar graph plane Preprint Proposition quantum groups quaternions recoupling theory regular isotopy regular isotopy invariant Reidemeister moves relation representation rotation shown solution space spin network strands string structure summation tangle Temperley-Lieb algebra Theorem tree trefoil tunnel link twist unknot unoriented Vassiliev invariants vector vertex weights vertices Wilson loop Witten Yang-Baxter Equation zero