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This paper develops a new chain model for the commutative graph complex$\mathsf{GC}_2$ which takes Lie graph homology as an input. Our main technicalresult is the identification of a large contractible complex of (certain)tadpoles and higher genus vertices of the Feynman transform of Lie graphhomology. Using this result we identify the anti-invarints of Lie graphhomology in genus $2$ with relations between bracketings of conjecturalgenerators of $\mathfrak{grt}_1$ in depth 2 modulo depth 3, unifying two apriori disparate appearances of the space of modular cusp forms in the study ofgraph homology.

Lie graph homology model for $\mathfrak{grt}_1$