Melvin-Morton conjecture and primitive Feynman diagrams

Abstract

We give a very short proof of the Melvin-Morton conjecture relating the colored Jones polynomial and the Alexander polynomial of knots. The proof is based on the explicit evaluation of the corresponding weight systems on primitive elements of the Hopf algebra of chord diagrams which, in turn, follows from simple identities between four-valent tensors on the Lie algebra $sl_2$ and the Lie superalgebra $gl(1|1)$. This shows that the miraculous connection between the Jones and Alexander invariants follows from the similarity (supersymmetry) between $sl_2$ and $gl(1|1)$.