Kontsevich graphs act on Nambu–Poisson brackets, II. The tetrahedral flow is a coboundary in 4D

Abstract Kontsevich constructed a map from suitable cocycles in the graph complex to infinitesimal deformations of Poisson bi-vector fields. Under the deformations, the bi-vector fields remain Poisson. We ask, are these deformations trivial, meaning, do they amount to a change of coordinates along a vector field? We examine this question for the tetrahedron, the smallest nontrivial suitable graph cocycle in the Kontsevich graph complex, and for the class of Nambu–Poisson brackets on ℝ d . Within Kontsevich’s graph calculus, we use dimension-specific micro-graphs, in which each vertex represents an ingredient of the Nambu–Poisson bracket. For the tetrahedron, Kontsevich knew that the deformation is trivial for d = 2 (1996). In 2020, Buring and the third author found that the deformation is trivial for d = 3. Building on these discoveries, we now establish that the deformation is trivial for d = 4.