This volume provides an introduction to knot and link invariants as generalized amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally includes a 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 1 is a systematic course on knots and physics starting from the ground up; and Part 2 is a set of lectures on various topics related to Part 1. Part 2 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.
Author(s): Louis H. Kauffman
Series: K & E Series on Knots and Everything
Edition: 3 Sub
Publisher: World Scientific Publishing Company
Year: 2001
Language: English
Pages: 788
Table of Contents......Page 16
Preface to the First Edition......Page 8
Preface to the Second Edition......Page 12
Preface to the Third Edition......Page 14
PART I. A SHORT COURSE OF KNOTS AND PHYSICS.......Page 20
1°. Physical Knots.......Page 21
2°. Diagrams and Moves.......Page 25
3°. States and the Bracket Polynomial.......Page 42
4°. Alternating Links and Checkerboard Surfaces.......Page 56
5°. The Jones Polynomial and its Generalizations.......Page 66
6°. An Oriented State Model for VK(t).......Page 91
7°. Braids and the Jones Polynomial.......Page 102
8°. Abstract Tensors and the Yang-Baxter Equation.......Page 121
9°. Formal Feynman Diagrams, Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)q.......Page 134
10°. The Form of the Universal R-matrix.......Page 165
11°. Yang-Baxter Models for Specializations of the Homfly Polynomial.......Page 178
12°. The Alexander Polynomial.......Page 191
13°. Knot-Crystals - Classical Knot Theory in Modern Guise.......Page 203
14°. The Kauffman Polynomial.......Page 232
15°. Oriented Models and Piecewise Linear Models.......Page 252
16°. Three Manifold Invariants from the Jones Polynomial.......Page 267
17°. Integral Heuristics and Witten’s Invariants.......Page 302
18°. Appendix - Solutions to the Yang-Baxter Equation.......Page 333
1°. Theory of Hitches.......Page 340
2°. The Rubber Band and Twisted Tube.......Page 346
3°. On a Crossing.......Page 349
4°. Slide Equivalence.......Page 353
5°. Unoriented Diagrams and Linking Numbers.......Page 356
6°. The Penrose Chromatic Recursion.......Page 363
7°. The Chromatic Polynomial.......Page 370
8°. The Potts Model and the Dichromatic Polynomial.......Page 381
9°. Preliminaries for Quantum Mechanics, Spin Networks and Angular Momentum.......Page 398
10°. Quaternions, Cayley Numbers and the Belt Trick.......Page 420
11°. The Quaternion Demonstrator.......Page 444
12°. The Penrose Theory of Spin Networks.......Page 460
13°. Q-Spin Networks and the Magic Weave.......Page 476
14°. Knots and Strings - Knotted Strings.......Page 492
15°. DNA and Quantum Field Theory.......Page 505
16°. Knots in Dynamical Systems - The Lorenz Attractor.......Page 518
CODA.......Page 528
REFERENCES......Page 530
Index......Page 548
APPENDIX......Page 556
Second Article......Page 558
Third Article......Page 559
Fourth Article......Page 560
I. Introduction......Page 568
II. Knots and the Gauss Code......Page 570
III. Jordan Curves and Immersed Plane Curves......Page 576
IV. The Abstract Tensor Model for Link Invariants......Page 581
V. From Abstract Tensors to Quantum Algebras......Page 588
VI. From Quantum Algebra to Quantum Groups......Page 599
VII. Categories......Page 605
VIII. Invariants of 3-Manifolds......Page 608
IX. Epilogue......Page 611
I. Introduction......Page 614
II. Trees and Four Colors......Page 615
III. The Temperley Lieb Algebra......Page 620
IV. Temperley Lieb Recoupling Theory......Page 628
V. Penrose Spin Networks......Page 630
VI. Knots and 3-Manifolds......Page 640
VII. The Shadow World......Page 644
VIII. The Invariants of Ooguri, Crane and Yetter......Page 650
LINK POLYNOMIALS AND A GRAPHICAL CALCULUS......Page 655
0. Introduction......Page 656
1. Rigid Vertex Isotopy......Page 657
2. The Homfly Polynomial......Page 666
3. Braids and the Hecke Algebra......Page 676
4. Demonstration of Identities in Oriented Graphical Calculus......Page 680
5. The Dubrovnik Polynomial......Page 684
Knots, Tangles, and Electrical Networks......Page 701
1. INTRODUCTION......Page 702
2. KNOTS, TANGLES, AND GRAPHS......Page 703
3. CLASSICAL ELECTRICITY......Page 712
4. MODERN ELECTRICITY - THE CONDUCTANCE INVARIANT......Page 714
5. TOPOLOGY: MIRROR IMAGES, TANGLES AND CONTINUED FRACTIONS......Page 725
6. CLASSICAL TOPOLOGY......Page 730
1 Introduction......Page 741
2 Vassiliev Invariants and Invariants of Rigid Vertex Graphs......Page 742
3 Vassiliev Invariants and Witten's Functional Integral......Page 745
4 Gaussian Integrals......Page 763
5 The Three-Dimensional Perturbative Expansion......Page 766
6 Wilson Lines, Light-Cone Gauge and the Kontsevich Integrals......Page 771
7 Formal Integration......Page 779
