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Knots and Physics 3rd Edition by Louis H Kauffman ISBN 0585459177 9789810241124

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Author(s): Louis H. Kauffman
ISBN(s): 9789810241124, 0585459177
Edition: 3 Sub
File Details: PDF, 20.05 MB
Year: 2001
Language: english
SKU: EB-1526970 Category: Tag:

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ISBN 10: 0585459177 
ISBN 13: 9789810241124
Author: Louis H Kauffman

This 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.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 “Knot Theory and Functional Integration” 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.

Knots and Physics 3rd Table of contents:

PART I. 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 Homfly Polynomial.
12°. The Alexander Polynomial.
13°. Knot-Crystals – Classical Knot Theory in 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.
PART II. 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
First Article
Second Article
Third Article
Fourth Article
Gauss Codes, Quantum Groups and Ribbon Hopf Algebras
I. Introduction
II. Knots and the Gauss Code
III. Jordan Curves and Immersed Plane Curves
IV. The Abstract Tensor Model for Link Invariants
V. From Abstract Tensors to Quantum Algebras
VI. From Quantum Algebra to Quantum Groups
VII. Categories
VIII. Invariants of 3-Manifolds
IX. Epilogue
Spin Networks, Topology and Discrete Physics
I. Introduction
II. Trees and Four Colors
III. The Temperley Lieb Algebra
IV. Temperley Lieb Recoupling Theory
V. Penrose Spin Networks
VI. Knots and 3-Manifolds
VII. The Shadow World
VIII. The Invariants of Ooguri, Crane and Yetter
LINK POLYNOMIALS AND A GRAPHICAL CALCULUS
0. Introduction
1. Rigid Vertex Isotopy
2. The Homfly Polynomial
3. Braids and the Hecke Algebra
4. Demonstration of Identities in Oriented Graphical Calculus
5. The Dubrovnik Polynomial
Knots, Tangles, and Electrical Networks
1. INTRODUCTION
2. KNOTS, TANGLES, AND GRAPHS
3. CLASSICAL ELECTRICITY
4. MODERN ELECTRICITY – THE CONDUCTANCE INVARIANT
5. TOPOLOGY: MIRROR IMAGES, TANGLES AND CONTINUED FRACTIONS
6. CLASSICAL TOPOLOGY
Knot Theory and Functional Integration
1 Introduction
2 Vassiliev Invariants and Invariants of Rigid Vertex Graphs
3 Vassiliev Invariants and Witten’s Functional Integral
4 Gaussian Integrals
5 The Three-Dimensional Perturbative Expansion
6 Wilson Lines, Light-Cone Gauge and the Kontsevich Integrals
7 Formal Integration

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