Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus

Let $P$ be a Poisson structure on a finite-dimensional affine real manifold. Can $P$ be deformed in such a way that it stays Poisson? The language of Kontsevich graphs provides a universal approach – with respect to all affine Poisson manifolds – to finding a class of solutions to this deformation problem. For that reasoning, several types of graphs are needed. In this paper we outline the algorithms to generate those graphs. The graphs that encode deformations are classified by the number of internal vertices $k$; for $k \leqslant 4$wepresentallsolutionsofthedeformationproblem\mathrm{.}For$k \geqslant 5$,firstreproducingthepentagon-wheelpicturesuggestedat$k=6$ by Kontsevich and Willwacher, we construct the heptagon-wheel cocycle that yields a new unique solution without $2$-loops and tadpoles at $k=8$.