Title: The heptagon-wheel cocycle in the Kontsevich graph complex,
Abstract: 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
$(2\ell+1)$-wheel graph with a nonzero coefficient at every
$\ell\in\mathbb{N}$. We present detailed calculations of the differential of
graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at $\ell =
1$ and $\ell = 2$ of one and two graphs respectively, the cocycle condition
$d(\gamma) = 0$ is verified by hand. For the next, heptagon-wheel cocycle
(known to exist at $\ell = 3$), we provide an explicit representative: it
consists of 46 graphs on 8 vertices and 14 edges.