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 %2b (2⌊m/2⌋) %2b (2⌈m/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 (2⌊m/2⌋ %2b (2⌈m/2⌉). The proof is based on Riskin%27s result that the Cartesian product C3 × C5 cannot be embedded in the Klein bottle. © 2003 Elsevier Science (USA). All rights reserved.
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 %2b (2⌊m/2⌋) %2b (2⌈m/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 (2⌊m/2⌋ %2b (2⌈m/2⌉). The proof is based on Riskin's result that the Cartesian product C3 × C5 cannot be embedded in the Klein bottle. © 2003 Elsevier Science (USA). All rights reserved.

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