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Our main result is that a 1971 conjecture due to Paul Kainen is false. Kainen's conjecture implies that the genus 2 crossing number of K 9 is 3. We disprove the conjecture by showing that the actual value is 4. The method used is a new one in the study of crossing numbers, involving proof of the impossibility of certain genus 2 embeddings of Ks.
Riskin, A. (1995). The genus 2 crossing number of K9. Discrete Mathematics, 145(1), 211–227. https://doi.org/10.1016/0012-365X(94)00037-J