This invaluable book is 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 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 new edition, articles on other topics, including Khovanov Homology, have been included.

Description-Table Of Contents

pt. 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 a 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 -- pt. 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 wave. 14. Knots and strings - knotted strings. 15. DNA and quantum field theory. 16. Knots in dynamical systems - the Leronz attractor