Knots and physics / Louis H. Kauffman.

Format:

Book

Author/Creator:

Kauffman, Louis H., 1945-

Contributor:

Alumni and Friends Memorial Book Fund.

Series:

K & E series on knots and everything ; v. 1.

K & E series on knots and everything ; v. 1

Language:

English

Subjects (All):

Knot polynomials.

Mathematical physics.

Physical Description:

xvi, 770 pages : illustrations ; 22 cm.

Edition:

Third edition.

Place of Publication:

Singapore ; River Edge, NJ : World Scientific, [2001]

Summary:

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.

Contents:

Part I. A Short Course of Knots and Physics

1. Physical Knots 4

2. Diagrams and Moves 8

3. States and the Bracket Polynomial 25

4. Alternating Links and Checkerboard Surfaces 39

5. The Jones Polynomial and its Generalizations 49

6. An Oriented State Model for V[subscript K](t) 74

7. Braids and the Jones Polynomial 85

8. Abstract Tensors and the Yang-Baxter Equation 104

9. Formal Feynman Diagrams, Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)[subscript q] 117

10. The Form of the Universal R-matrix 148

11. Yang-Baxter Models for Specializations of the Homfly Polynomial 161

12. The Alexander Polynomial 174

13. Knot-Crystals - Classical Knot Theory in a Modern Guise 186

14. The Kauffman Polynomial 215

15. Oriented Models and Piecewise Linear Models 235

16. Three Manifold Invariants from the Jones Polynomial 250

17. Integral Heuristics and Witten's Invariants 285

18. Appendix - Solutions to the Yang-Baxter Equation 316

Part II. Knots and Physics

Miscellany

1. Theory of Hitches 323

2. The Rubber Band and Twisted Tube 329

3. On a Crossing 332

4. Slide Equivalence 336

5. Unoriented Diagrams and Linking Numbers 339

6. The Penrose Chromatic Recursion 346

7. The Chromatic Polynomial 353

8. The Potts Model and the Dichromatic Polynomial 364

9. Preliminaries for Quantum Mechanics, Spin Networks and Angular Momentum 381

10. Quaternions, Cayley Numbers and the Belt Trick 403

11. The Quaternion Demonstrator 427

12. The Penrose Theory of Spin Networks 443

13. Q-Spin Networks and the Magic Weave 459

14. Knots and Strings - Knotted Strings 475

15. DNA and Quantum Field Theory 488

16. Knots in Dynamical Systems - The Lorenz Attractor 501

Gauss Codes, Quantum Groups and Ribbon Hopf Algebras 551

Spin Networks, Topology and Discrete Physics 597

Link Polynomials and a Graphical Calculus / P. Vogel 638

Knots, Tangles, and Electrical Networks / J. R. Goldman 684

Knot Theory and Functional Integration 724.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

9810241127

OCLC:

48475562