Bad drawings of small complete graphs

Abstract

We show that for K 5 (resp. K 3,3 ) there is a drawing with i independent crossings, and no pair of independent edges cross more than once, provided i is odd with 1 ≤ i ≤ 15 (resp. 1 ≤ i ≤ 17). Conversely, using the deleted product cohomology, we show that for K 5 and K 3,3 , if A is any set of pairs of independent edges, and A has odd cardinality, then there is a drawing in the plane for which each element in A cross an odd number of times, while each pair of independent edges not in A cross an even number of times. For K 6 we show that there is a drawing with i independent crossings, and no pair of independent edges cross more than once, if and only if 3 ≤ i ≤ 40.