CBMS Regional Conference Series in Mathematics 1991; 113 pp; softcover Number: 80 Reprint/Revision History: third printing 2000 ISBN10: 0821807293 ISBN13: 9780821807293 List Price: US$25 Member Price: US$20 All Individuals: US$20 Order Code: CBMS/80
 This book is based on a set of lectures presented by the author at the NSFCBMS Regional Conference, Applications of Operator Algebras to Knot Theory and Mathematical Physics, held at the U.S. Naval Academy in Annapolis in June 1988. The audience consisted of lowdimensional topologists and operator algebraists, so the speaker attempted to make the material comprehensible to both groups. He provides an extensive introduction to the theory of von Neumann algebras and to knot theory and braid groups. The presentation follows the historical development of the theory of subfactors and the ensuing applications to knot theory, including full proofs of some of the major results. The author treats in detail the Homfly and Kauffman polynomials, introduces statistical mechanical methods on knot diagrams, and attempts an analogy with conformal field theory. Written by one of the foremost mathematicians of the day, this book will give readers an appreciation of the unexpected interconnections between different parts of mathematics and physics. Reviews "This book is a set of lectures by the author on von Neumann algebras, braid groups and invariants of knots and links. Since the author is the person who discovered (in 1984) the extraordinary connections among these subjects, it is delightful to have an exposition of these ideas from him ... a unique resource."  Mathematical Reviews "Even a superficial perusal of the book will teach something. Belongs on every mathematician's shelf."  The Bulletin of Mathematics Books and Computer Software "The author includes an introduction, in his refreshing and very clear style, to the basics of each of these subjects, making these notes very readable also for nonspecialists ... These notes are a must for anybody who wants to familiarize himself with this fascinating circle of ideas. But also for the expert they give new insights and present many ideas more clearly than the original articles."  Zentralblatt MATH "Although these lectures were delivered 9 years ago and their subject matter has undergone a very rapid development since then, their informal style, freshness of presentation and unique personal insights on the surprising interconnections between apparently different topics in mathematics and mathematical physics continue to make them compulsory and highly rewarding reading. No reader wishing to gain insight into one of the most exciting mathematical developments of the past 15 years could do better than go straight to these lectures by the master (VFRJ) for inspiration."  Monatshefte für Mathematik Table of Contents  von Neumann algebras
 Group actions and subfactors
 Values of the index, Virasoro algebra
 Construction of examples, further structure
 The braid group and its representations
 Knots and links
 The knot polynomial \(V_L\)
 Knots and statistical mechanics
 The algebraic approach
