The ($sl(N,\mathbb{C})$,ad)-weight system is a linear map from the vector space generated by chord diagrams to $\mathbb{C}$ associated with simple Lie algebra $sl(N,\mathbb{C})$ and its adjoint representation ad. This linear map derives the universality of Kontsevich invariant to the quantum invariants and Vassiliev invariants of links.

In this talk, I derive a formula of the ($sl(N,\mathbb{C})$,{\rm ad})-weight system for chord diagram. This is useful for quick calculation of weight system. Furthermore this formula is a generalization of the formulae obtained by Chmutov-Varchenko and Kuga-Yoshizumi in the case of ($sl(2,\mathbb{C})$,ad)