Vassiliev invariants and the Hopf algebra of chord diagrams

This paper is closely related to Bar-Natan's work, and fills in some of the gaps in [1]. Folio-wing his analogy of the extension of knot invariants to knots with double points to the notion of multivariate calculus on polynomials, we introduce a new notation which facilitates the formulation of a Leibniz type formula for the product of two Vassiliev invariants. This leads us to see how Bar-Natan's co-product of chord diagrams corresponds to multiplication of Vassiliev invariants. We also include a proof that the multiplication in A is a consequence of Bar-Natan's 4T relation. The last part of this paper consists of a proof that the space of weight systems isa sub-Hopf algebra of the space A*, by means of the canonical projection.