## Knots and PhysicsThis 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 a 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. The book is divided into two parts: Part I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II includes topics such as frictional properties of knots, relations with combinatorics, and knots in dynamical systems. In this third edition, a paper by the author entitled "Functional Integration and Vassiliev invariants" has been added. This paper shows how the Kontsevich integral approach to the Vassiliev invariants is directly related to the perturbative expansion of Witten's functional integral. While the book supplies the background, this paper can be read independently as an introduction to quantum field theory and knot invariants and their relation to quantum gravity. As in the second edition, there is a selection of papers by the author at the end of the book. Numerous clarifying remarks have been added to the text. |

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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 flmatrix | 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 |