The heptagon-wheel cocycle in the Kontsevich graph complex. (21st December 2017) - Buring, Ricardo; Kiselev, Arthemy V.; Rutten, Nina J.
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 2 n − 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 (2ℓ + 1)-wheel graph with a nonzero coefficient at every ℓ∈ℕ. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel cocycles, consisting at ℓ = 1 and ℓ = 2 of one and two graphs respectively, the cocycle condition d( γ ) = 0 is verified by hand. For the next, heptagonwheel cocycle (known to exist at ℓ = 3), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.