Kontsevich graphs act on Nambu-Poisson brackets, III. Uniqueness aspects

Kontsevich constructed a map between `good’ graph cocycles $\gamma$ and infinitesimal deformations of Poisson bivectors on affine manifolds, that is, Poisson cocycles in the second Lichnerowicz–Poisson cohomology. For the tetrahedral graph cocycle ${\gamma }_3$ and for the class of Nambu-determinant Poisson bivectors $P$ over ${\mathbb{R}}^2$, ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$, we know the fact of trivialization, $\dot{P}=[[P,{\vec{X}}_{\text{dim}}^{{\gamma }_3}]]$, by using dimension-dependent vector fields ${\vec{X}}_{\text{dim}}^{{\gamma }_3}$ expressed by Kontsevich (micro-) graphs. We establish that these trivializing vector fields ${\vec{X}}_{\text{dim}}^{{\gamma }_3}$ are unique modulo Hamiltonian vector fields ${\vec{X}}_H=d_P(H)=[[P,H]]$, where $d_P$ is the Lichnerowicz–Poisson differential and where the Hamiltonians $H$ are also represented by Kontsevich (micro-)graphs. However, we find that the choice of Kontsevich (micro-)graphs to represent the aforementioned multivectors is not unique.