  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 $1\le i\le 15$ (resp.~ $1\le i\le 17$). 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 $3\le i\le 40$.
