A proof of the Melvin-Morton conjecture and Feynman diagrams

The Melvin-Morton conjecture says how the Alexander-Conway knot invariant function can be read from the coloured Jones function. It has been proved by D. Bar-Natan and S. Garoufalidis. They reduced the conjecture to a statement about weight systems. The proof of the latter is the most difficult part of their paper. We give a new proof of the statement based on the Feynman diagram description of the primitive space of the Hopf algebra A of chord diagrams.