The heptagon-wheel cocycle in the Kontsevich graph complex

TL;DR

This paper investigates the structure of cocycles in the Kontsevich graph complex, providing explicit calculations and a detailed example of a heptagon-wheel cocycle with 46 graphs, advancing understanding of graph-based symmetries in Poisson geometry.

Contribution

It explicitly constructs and verifies a heptagon-wheel cocycle in the Kontsevich graph complex, extending known results for smaller wheels.

Findings

Explicit calculation of differential for tetrahedron and pentagon-wheel cocycles

Verification of cocycle condition by hand for these cases

Construction of a heptagon-wheel cocycle with 46 graphs on 8 vertices and 14 edges

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.