by Louis H Kauffman (University of Illinois, Chicago)

In this second edition, the following recent papers have been added: “Gauss Codes, Quantum Groups and Ribbon Hopf Algebras”, “Spin Networks, Topology and Discrete Physics”, “Link Polynomials and a Graphical Calculus” and “Knots Tangles and Electrical Networks”. An appendix with a discussion on invariants of embedded graphs and Vassiliev invariants has also been included.

This book is an introduction to knot and link invariants as generalized amplitudes (vacuum–vacuum 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 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 2 parts: Part I of the book is a systematic course in knots and physics starting from the ground up. Part II is a set of lectures on various topics related to and sometimes based on Part I. Part II also explores some side-topics such as frictional properties of knots, relations with combinatorics and knots in dynamical systems.

• Physical Knots
• States and the Bracket Polynomial
• The Jones Polynomial and Its Generalizations
• Braids and the Jones Polynomial
• Formal Feynman Diagrams, Bracket as a Vacuum-Vacuum Expectation and the Quantum Group SL(2)q
• Yang-Baxter Models for Specializations of the Homfly Polynomial
• The Alexander Polynomial
• Knot-Crystals — Classical Knot Theory in Modern Guise
• The Kauffman Polynomial
• Three Manifold Invariants from the Jones Polynomial
• Integral Heuristics and Witten' s Invariants
• The Chromatic Polynomial
• 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
  • and other papers