Drawings of complete graphs in the projective plane

Hill's Conjecture states that the crossing number cr(𝐾𝑛) of the complete graph 𝐾𝑛 in the plane (equivalently, the sphere) is 14⌊𝑛2⌋⌊𝑛−12⌋⌊𝑛−22⌋⌊𝑛−32⌋=𝑛4/64+𝑂(𝑛3) . Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely 𝑛4/64+𝑂(𝑛3) , thus matching asymptotically the conjectured value of cr(𝐾𝑛) . Let cr𝑃(𝐺) denote the crossing number of a graph 𝐺 in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of 𝐾𝑛 is (𝑛4/8𝜋2)+𝑂(𝑛3) . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if lim𝑛→∞ cr𝑃(𝐾𝑛)/𝑛4=1/8𝜋2 . We construct drawings of 𝐾𝑛 in the projective plane that disprove this.