This book is an introduction to knot and link invariants as generalized amplitudes (vacuum-vacuum amplitudes) for a quasi-physical process. The demands of the knot theory, coupled with a quantum statistical framework, create a context that naturally includes a wide range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward the knot theory and its relations with these subjects. This has the advantage of providing very direct access to the algebra and to the combinatorial topology, as well as the physical ideas. This book is divided into two parts: Part one is a systematic course in knots and physics starting from the ground up. Part two is a set of lectures on various topics related with and sometimes based on part one. It also explores some side-topies such as frictional properties of knots, relations with combinatorics, knots and dynamical systems. In this second edition, an appendix has been added with a discussion of invariants of embedded graphs and Vassiliev invariants.
Included is a reprinting of three recent papers by the author on quantum groups and invariants of 3-manifolds, spin networks and graph invariants.
Physical Knots; States and the Bracket Polynomial; The Jones Polynominal and Its Generalizations; Braids and Polynomials: Formal Feynman Diagrams, Bracket as Vacuum-Vacmum expectation and the Quantum Group SL(2)q; Yang-Baxter Models for Specialization's of the Homfly Polynomial; The Alexander Polynomial; Knot Crystals - Classical Knot Theory in Modem Guise; The Kauffman Polynomial; Three-Manifold Invariants from the Jones Polynomials; integral Heuristics and Witten's lnvariants; Chromatic Polynomials; The Potts Model and the Dichromatic Polynomial; The Penrose Theory of Spin Networks; Knots and Strings - Knotted Strings; DNA and Quantum Field Theory; Knots in Dynamical Systems - The Lorenz Attractor.