Szücs Kauffman ’ s state model for the Jones polynomial

Introduction: In recent years a number of fundamental ideas and methods of mathematical physics have penetrated through psychological barriers between physics and topology. In the knot theory this development was initiated by V. Jones who used von Neumann algebras to construct a new polynomial invariant of knots and links in the 3-dimensional sphere S 3. The Jones's discovery gave impetus to an enormous development in the knot theory, 3-dimensional topology and related domains. This development was the main subject of the meeting. Since 1984 when Jones introduced his polynomial the main line of attack was explanation of the nature of this polynomial and its generalization, say, to links in other 3-manifolds. To the moment of writing, several different points of view on the Jones polynomial have been developed basing on various techniques coming from algebra and mathematical physics. Here is a short but impressive list of theories more or less directly involved in the subject: theory of quantum groups, conformal field theory in dimension 2, representation theory of symmetric groups and Hecke algebras, theory of exactly solvable models of statistical mechanics etc.. The Arbeitsgemeinschaft considered 4 different though related lines of study forming the main body of the theory. I. The first and most algebraic approach stems directly from the original Jones's paper. It involves Temperley-Lieb algebras, Hecke algebras and their natural modifications due to Birman-Wenzl. The core of this approach is the theory of braid groups and their linear representations. II. The second approach was concerned with Witten's ideas, relating the Jones polynomial to the conformal field theories in dimension 2 and Chern-Simons invariants. From the topolog-ical viewpoint the important achievement of Witten is the inclusion of the Jones polynomial in a more general picture of topological quantum field theories. III. The approach based on the statistical mechanical models and the theory of quantum groups: A state sum model for the Jones polynomial was introduced by Kauffman. For other related polynomials one involves more general vertex models associated with R-matrices. This line is crowned with a construction of the topological quantum field theory in dimension 3 extending the Jones polynomial (and predicted by Witten).