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In recent work of Cachazo, Guevara, Mizera and the author, a generalizationof the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ wasintroduced as an integral over the moduli space of $n$ points in$\mathbb{CP}^{k-1}$, with value a sum of certain rational functions on thekinematic space $\mathcal{K}_{k,n}$. It was shown there for$m^{(3)}(\mathbb{I}_6,\mathbb{I}_6)$ and later by Cachazo and Rojas thatcollections of poles appearing in $m^{(3)}(\mathbb{I}_7,\mathbb{I}_7)$ arecompatible exactly when they are dual to collections of rays which generate themaximal faces of a polyhedral complex known as the (nonnegative) tropicalGrassmannian.  In this note, we derive a remarkable planar basis for the space ofgeneralized kinematic invariants which coincides in the case $k=2$ with usualstandard planar multi-particle basis for the kinematic space. We implement inMathematica the action on formal linear combinations of planar matroidsubdivisions of a boundary operator which, together with the planar basis,determines compatibility for any given poles appearing in the expansion of$m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$, by computing a certain combinatorialnon-crossing condition on the second hypersimplicial faces $\Delta_{2,n-(k-2)}$of $\Delta_{k,n}$. The algorithms are implemented in an accompanyingMathematica notebook and are evaluated on existing tables of rays, in the formof tropical Plucker vectors, to tabulate the finest planar subdivisions of$\Delta_{3,8},\Delta_{3,9}$ and $ \Delta_{4,8}$, or equivalently the set ofmaximal cones for the corresponding nonnegative tropical Grassmannians.

Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams