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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, thenthere is a drawing in the plane for which each element in $A$ cross an oddnumber of times, while each pair of independent edges not in $A$ cross an evennumber of times. For $K_6$ we show that there is a drawing with $i$ independentcrossings, and no pair of independent edges cross more than once, if and onlyif $3\le i\le 40$.

Bad drawings of small complete graphs