by Louis H Kauffman (University of Illinois, Chicago)

Kauffman was born in 1945. He graduated as valedictorian of his class at Norwood Norfolk Central High School in 1962. He received his B.S. at MIT in 1966 and his Ph.D. in mathematics from Princeton University in 1972. Kauffman has been a prominent leader in Knot Theory, one of the most active research areas in mathematics today. His discoveries include a state sum model for the Alexander-Conway Polynomial, the bracket state sum model for the Jones polynomial, the Kauffman polyomial and Virtual Knot Theory. He is the Editor-in-Chief of JKTR, Editor of the Series on Knots and Everything, full professor at UIC and author of numerous books related to the theory of knots — including “Knots and Physics”, “Knots and Applications”, “On Knots”, “Formal Knot Theory”.
 

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 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 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.

• 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
• Knot-Crystals — Classical Knot Theory in a 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 selected papers