The heptagon-wheel cocycle in the Kontsevich graph complex

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on $n$ vertices and $2n-2$ edges, induce – under the orientation mapping – infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial cocycle that contains the $(2\mathrm{\ell }+1)$-wheel graph with a nonzero coefficient at every $\mathrm{\ell }\in \mathbb{N}$. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at $\ell = 1$and$\ell = 2$ofoneandtwographsrespectively,thecocyclecondition$d(\gamma) = 0$ is verified by hand. For the next, heptagon-wheel cocycle (known to exist at $\mathrm{\ell }=3$), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.