ABSTRACT
When two edges cross twice in a drawing, we can swap the arcs involved, and reduce the number of crossings along both edges by two (we first encountered this move in Lemma 1.3). It seems as if we can think of crossings along an edge modulo 2. This thought inspires a different way to count crossings: the odd crossing number, ocr(D), of a drawing D is the number of pairs of edges that cross an odd number of times (oddly, from now on). When distinguishing between two edges crossing evenly or oddly, we call this the crossing parity of the two edges. The odd crossing number of a graph G, ocr(G), is the smallest ocr(D) of any drawing D of G. By the argument we sketched we should have ocr(G) = cr(G). We will see presently that this is not true (what is wrong with our intuitive argument that we can remove two crossings from a pair of edges?), but some of the intuition can be salvaged.
