27 Sep 2002

 Sergei Duzhin (St.-Petersburg Branch of the Steklov Mathematical Institute; Independent University of Moscow) 
"Weight Systems, Four-Colour Theorem and Skew-Symmetric Decomposable Functions"

The subject of this talk originates from topology (finite type knot invariants), but it is also related to classical combinatorics (four-colour theorem) and fairly elementary algebra (skew-symmetric functions). Weight systems are crucial in the combinatorial study of finite type (Vassiliev) knot invariants. They are defined as functions on the spaces generated by graphs with all vertices of valency 1 or 3, satisfying certain relations. 

There is a well-known construction of weight systems by metrized Lie algebras. We introduce a new family of weight systems in a sense dual to the family of Lie algebra weight systems. The basic component in our construction is a skew-symmetric function of three variables $f(x,y,z)$ that satisfies the following equation (we call it the Klein equation): $f(x,y,z)f(u,v,z)-f(x,u,z)f(y,v,z)+f(x,v,z)f(y,u,z)=0$, which is a counterpart of the Jacobi identity for the Lie algebra structure tensor.

The simplest skew-symmetric function in 3 variables is the polynomial $f=(x-y)(y-z)(z-x)$. We explain the relation of the corresponding weight system with the four-colour theorem (after Yu.Matiyasevich). Finally, we study the multivariate analog of the Klein equation. A skew-symmetric function in several variables $f(x_1,…,x_n)$ is said to be decomposable if it can be represented as a determinant $f=\det\{f_i(x_j)\}_{n\times n}$, where $f_i(x)$ are univariate functions. We prove a criterion of decomposability for analytic skew-symmetric functions in an arbitrary number of variables.

NB: No special knowledge is required. Students are welcomed.