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, $\dot{P}=[[ P, \vec{X}^{\gamma_3}_{\text{dim}}]]$, by using dimension-dependent vector fields $\vec{X}^{\gamma_3}_{\text{dim}}$ expressed by Kontsevich (micro-) graphs. We establish that these trivializing vector fields $\vec{X}^{\gamma_3}_{\text{dim}}$ 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.