3-symmetric and 3-decomposable geometric drawings of Kn

Even the most superficial glance at the vast majority of crossing-minimal geometric drawings of KnKn reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply 3-symmetric). And second, they all are 3-decomposable , that is, there is a triangle TT enclosing the drawing, and a balanced partition A,B,CA,B,C of the underlying set of points PP, such that the orthogonal projections of PP onto the sides of TT show AA between BB and CC on one side, BB between AA and CC on another side, and CC between AA and BB on the third side. In fact, we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric optimal constructions for all nn multiples of 3. In this paper, we show that any 3-decomposable geometric drawing of KnKn has at least 0.380029(n4)+Θ(n3) crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the general upper bound for the rectilinear crossing number of KnKn to 0.380488(n4)+Θ(n3). We also give explicit 3-symmetric and 3-decomposable constructions for n<100n<100 that are at least as good as those previously known.