Knots and statistical mechanics
The object of the paper is the consideration of the deep connection between the partition function (for example, in statistical mechanics and discrete networks) and the quantum invariants of knots and links. The author begins his review with the consideration of the models involving discrete networks. Then he recalls the definition of the scattering matrix in quantum and statistical mechanics. As examples he considers the coloring problem of 3-regular graphs in the plane, coloring in connection with the Potts model in quantum mechanics. Some combinatorial problems are touched. The next section is devoted to the well-known definition of the bracket polynomial. Further, the author describes the construction of quantum link invariants of knot and link diagrams that are arranged with respect to a given direction in the plane. The destination of the special directions as time allows the interpretation of the consequent evaluation of the diagram as a generalized scattering amplitude. The last section is devoted to a short description of the quantum link invariant which is built from an infinite sequence of interlocked Vassiliev invariants. An open problem is stated: whether the class of finite type Vassiliev invariants is greater than the one generated from quantum link invariants. In conclusion, some attention is given to unsolved problems in quantum field theory which are connected with the scattering matrix, i.e. the functional integral which have been of key importance for motivating the construction of new invariants of knots and three-manifolds and in elucidating the properties of Vassiliev invariants.
scientific article; zbMATH DE number 1370404
scientific article