紐結和物理學

內(nèi)容概要

This book has its origins in two short courses given by the author in Bologna and Torino, Italy during the Fall of 1985. At that time, connections between statistical physics and the Jones polynomial were just beginning to appear, and it seemed to be a good idea to write a book of lecture notes entitled Knots and Physics. The subject of knot polynomials was opening up, with the Jones polynomial as the first link polynomial able to distinguish knots from their mirror images. We were looking at the tip of an iceberg,t The field has grown by leaps and bounds with remarkable contributions from mathematicians and physicists - a wonderful interdisciplinary interplay. In writing this book I wanted to preserve the flavor of those old Bologna/Torino notes, and I wanted to provide a pathway into the more recent events. After a good deal of exploration, I decided, in 1989, to design a book divided into two parts. The first part would be combinatorial, elementary, devoted to the bracket polyno- mial as state model, partition function, vacuum-vacuum amplitude, Yang-Baxter model. The bracket also provides an entry point into the subject of quantum groups, and it is the beginning of a significant generalization of the Penrose spin- networks (see Part II, section 13.) Part II is an exposition of a set of related topics, and provides room for recent developments. In its first incarnation, Part II held material on the Potts model and on spin-networks.

書籍目錄

Table of Contents   Preface to the First Edition   Preface to the Second Edition   Preface to the Third Edition PartⅠ.A Short Course of Knots and Physics   1.Physical Knots   2.Diagrams and Moves   3.States and the Bracket Polynomial   4.Alternating Links and Checkerboard Surfaces   5.The Jones Polynomial and its Generalizations   6.An Oriented State Model for Vk(t)   7.Braids and the Jones Polynomial   8.Abstract Tensors and the Yang-Baxter Equation   9.Formal Feynman Diagrams， Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)q   10.The Form of the Universal R-matrix   11.Yang-Baxter Models for Specializations of the Homily Polynomial   12.The Alexander Polynomial   13.Knot-Crystals - Classical Knot Theory in a Modern Guise   14.The Kauffman Polynomial   15.Oriented Models and Piecewise Linear Models   16.Three Manifold Invariants from the Jones Polynomial    17.Integral Heuristics and Witten''''s Invariants   18.Appendix - Solutions to the Yang-Baxter Equation          PartⅡ.Knots and Physics - Miscellany   1.Theory of Hitches   2.The Rubber Band and Twisted Tube   3.On a Crossing                    4.Slide Equivalence                    5.Unoriented Diagrams and Linking Numbers    6.The Penrose Chromatic Recursion                     7.The Chromatic Polynomial                     8.The Potts Model and the Dichromatic Polynomial     9.Preliminaries for Quantum Mechanics, Spin Networks and Angular Momentum   10.Quaternions, Cayley Numbers and the Belt Trick   11.The Quaternion Demonstrator   12.The Penrose Theory of Spin Networks   13.Q-Spin Networks and the Magic Weave                   14.Knots and Strings - Knotted Strings                   15.DNA and Quantum Field Theory                 16.Knots in Dynamical Systems - The Lorenz Attractor           Coda                     References                     Index                   Appendix                     Introduction                     Gauss Codes, Quantum Groups and Ribbon Hopf Algebras    Spin Networks, Topology and Discrete Physics   Link Polynomials and a Graphical Calculus with P.Vogel     Knots, Tangles, and Electrical Networks with J.R.Goldman         Knot Theory and Functional Integration

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