Point sets that minimize (≤k)(≤k)-edges, 3-decomposable drawings, and the rectilinear crossing number of K30K30

There are two properties shared by all known crossing-minimizing geometric drawings of KnKn, for nn a multiple of 3. First, the underlying nn-point set of these drawings minimizes the number of (≤k)(≤k)-edges, that means, has exactly 3(k+22)(≤k)-edges, for all 0≤k