Knots and Physics

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World Scientific, 2001 - Science - 770 pages
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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 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
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