Kontsevich graphs act on Nambu--Poisson brackets, IV. When the invisible becomes crucial

Kontsevich’s graphs allow encoding multi-vectors whose coefficients are differential-polynomial in the coefficients of a given Poisson bracket on an affine real manifold. Encoding formulas by directed graphs adapts to the class of Nambu-determinant Poisson brackets, yet the graph topology becomes dimension-specific. To inspect whether a given Kontsevich graph cocycle $\gamma$ acts (non)trivially – in the second Poisson cohomology – on the space of Nambu brackets, taking a vector field solution ${\vec{X}}_d^{\gamma }$ from dimension $d$ does not work in $d+1$. For $2⩽d⩽4$, the action of tetrahedron ${\gamma }_3$ on Nambu brackets is known to be a Poisson coboundary, $\dot{P}=[[P,{\vec{X}}_d^{{\gamma }_3}(P)]]$. We explore which minimal (sub)sets of graphs, encoding (non)vanishing objects over ${\mathbb{R}}_{\text{aff}}^d$, generate the topological data that suffice for a solution ${\vec{X}}_{d+1}^{{\gamma }_3}$ to appear. We detect that there can be no solution in higher dimension without invisible graphs that vanish as formulas in $d=3$, but whose descendants do not all vanish over $d=4$.