Optimal meshes of curves in the Klein bottle

It is proved that if n is sufficiently large compared to m, and m⩾3, then the minimum number of intersections in an (m,n)-mesh of curves in the Klein bottle equals mn+(⌊m/2⌋2)+(⌈m/2⌉2). As a corollary, it follows that for each m⩾3 there is an N0(m) such that, for n⩾N0(m), the Klein bottle crossing number of Cm×Cn equals (⌊m/2⌋2)+(⌈m/2⌉2). The proof is based on Riskinʹs result that the Cartesian product C3×C5 cannot be embedded in the Klein bottle.