The first homology of ${\mathrm{IA}}_n$ with coefficients in spaces of Jacobi diagrams

The aim of this paper is to compute the first homology of ${\mathrm{IA}}_n$ with coefficients in the $\mathrm{Aut}(F_n)$-module $A_2(n)$ of Jacobi diagrams of degree $2$ on $n$-component oriented arcs. We also prove the self-duality of the $\mathrm{Aut}(F_3)$-submodule $A_2^{\mathrm{\prime }\mathrm{\prime }}(3)$ of $A_2(3)$ which is generated by a certain element.