  In recent work of Cachazo, Guevara, Mizera and the author, a generalization
of the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ was
introduced as an integral over the moduli space of $n$ points in
$\mathbb{CP}^{k-1}$, with value a sum of certain rational functions on the
kinematic space $\mathcal{K}_{k,n}$. It was shown there for
$m^{(3)}(\mathbb{I}_6,\mathbb{I}_6)$ and later by Cachazo and Rojas that
collections of poles appearing in $m^{(3)}(\mathbb{I}_7,\mathbb{I}_7)$ are
compatible exactly when they are dual to collections of rays which generate the
maximal faces of a polyhedral complex known as the (nonnegative) tropical
Grassmannian.
  In this note, we derive a remarkable planar basis for the space of
generalized kinematic invariants which coincides in the case $k=2$ with usual
standard planar multi-particle basis for the kinematic space. We implement in
Mathematica the action on formal linear combinations of planar matroid
subdivisions 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 combinatorial
non-crossing condition on the second hypersimplicial faces $\Delta_{2,n-(k-2)}$
of $\Delta_{k,n}$. The algorithms are implemented in an accompanying
Mathematica notebook and are evaluated on existing tables of rays, in the form
of tropical Plucker vectors, to tabulate the finest planar subdivisions of
$\Delta_{3,8},\Delta_{3,9}$ and $ \Delta_{4,8}$, or equivalently the set of
maximal cones for the corresponding nonnegative tropical Grassmannians.
