ABSTRACT
This chapter discusses the algebraic structure that arises on the set of Vassiliev knot invariants. The proof of the fact that the IHX-relation holds can be reduced to the case when one of the four vertices is an exterior one. This can be done by taking the lower left vertex for all diagrams that have to satisfy the IHX-relation and then splitting all interior vertices between this vertex and the circle in the same manner for all diagrams. The beautiful observation is that the STU-relation for Feynman diagrams represents the Jacobi identity for Lie algebras. Thus, the constructed numbers are indeed invariant under the STU-relation. This construction is the simplest case of the general construction; it deals only with adjoint representations of Lie algebras. There is a beautiful idea connecting the representation theory of Lie algebras, and knot theory.
