Knots and Physics

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World Scientific, 1991 - Mathematics - 538 pages
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The subject of knot polynomials has grown by leaps and bounds, with contributions from mathematicians and physicists. This book has its origins in two short courses given by the author in Italy in 1985. The first part is combinatorial, elementary, devoted to the bracket polynomial 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. Part II is an exposition of a set of related topics, and provides room for recent developments. Paper edition (unseen), $28. Annotation copyrighted by Book News, Inc., Portland, OR
  

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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
References
513
Index
531
Copyright

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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.

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