INTERSECTION OF CURVES AND CROSSING NUMBER OF C-M X C-N ON SURFACES

Let (K-1, K-2) be two families of closed curves on a surface S, such t hat \K-1\ = m, \K-2\ = n, m(0) less than or equal to m less than or eq ual to n, each curve in K-1 intersects each curve in K-2, and no point of S is covered three times. When S is the plane, the projective plan e, or the Klein bottle, we prove that the total number of intersection s in K-1 boolean OR K-2 is at least 10mn/9, 12mn/11, and mn + 10(-13)m (2), respectively. Moreover, when m is close to n, the constants are i mproved. For instance, the constant for the plane, 10/9, is improved t o 8/5, for n less than or equal to(m - 1)/4. Consequently, we prove lo wer bounds on the crossing number of the Cartesian product of two cycl es, in the plane, projective plane, and the Klein bottle. All lower bo unds are within small multiplicative factors from easily derived upper bounds. No general lower bound has been previously known, even on the plane.