## Knots and PhysicsThis book is an introductory explication on the theme of knot and link invariants as generalized amplitudes (vacuum-vacuum amplitudes) for a quasi-physical process. The demands of the knot theory, coupled with a quantum statistical frame work create a context that naturally and powerfully includes an extraordinary range of interelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward the knot theory and its relations with these subjects. This has the advantage of providing very direct access to the algebra and to the combinatorial topology, as well as the physical ideas. This book is divided into 2 parts: Part I of the book is a systematic course in knots and physics starting from the ground up. Part II is a set of lectures on various topics related with and sometimes based on Part I. Part II also explores some side-topics such as frictional properties of knots, relations with combinatorics, knots in dynamical systems. |

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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 Vxt | 74 |

Braids and the Jones Polynomial | 85 |

The Rubber Band and Twisted Tube | 329 |

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 |

Preliminaries for Quantum Mechanics Spin Networks and Angular Momentum | 381 |

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 J?matrix | 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 |

Quaternions Cayley Numbers and the Belt Trick | 403 |

The Quaternion Demonstrator | 427 |

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 |

Coda | 511 |

513 | |

531 | |

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### Common terms and phrases

abstract tensors Alexander polynomial ambient isotopy arcs associated assume belt trick bracket polynomial braid group calculation chromatic colors commutation completes the proof components configurations construction corresponding crossing crystal curve defined deformation denotes the number diagrammatic edge elements embedded example Figure follows formalism formula framing gauge given graph Hence Homfly polynomial identity index set indicated interaction isotopy invariant iZ-matrix Jones polynomial knot or link knot theory knots and links labelled Lemma Lie algebra link diagram linking number Lorentz matrix mirror image multiplication Note obtained oriented link planar plane Potts model Proposition quandle quantum group quaternions regular isotopy regular isotopy invariant Reidemeister moves relations Remark representation rotation satisfies shows solution space spacetime spin network splicing strands string structure summation tangle Temperley-Lieb algebra Theorem three-manifold topological trefoil twist type II move unknot unoriented vacuum-vacuum vector vertex weights vertices Wilson loop writhe Yang-Baxter Equation zero

### Popular passages

Page 516 - Reshetikhin, LA Takhtajan, Quantization of Lie Groups and Lie Algebras, LOMI preprint (1987).

Page 527 - B. Trace, On the Reidemeister moves of a classical knot, Proc. Amer. Math. Soc.

Page 513 - Virasoro algebra, von Neumann algebra and critical eight-vertex SOS models", J. Phys. Soc. Japan 5_5_, No. 10, 3285-3288 (1986). 92) Lawrence, R., "A universal link invariant using quantum groups

Page 515 - VG Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl.

Page 514 - Burgoyne PN 1963 Remarks on the combinatorial approach to the Ising problem J.

Page 514 - MF Atiyah. Geometry of Yang-Mills Fields. Accademia Nazionale dei Lincei Scuola Normale Superiore - Lezioni Fermiane. Pisa (1979). [BE] HJ Bernstein and AV Phillips. Fiber bundles and quantum theory.