The genus 2 crossing number of K9 Original Research Article

Our main result is that a 1971 conjecture due to Paul Kainen is false. Kainenʹs conjecture implies that the genus 2 crossing number of K9 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 K8.