On the crossing number of K₁3

Since the crossing number of K_{12} is now known to be 150, it is well-known that simple counting arguments and Kleitman's parity theorem for the crossing number of K_{2n+1} combine with a specific drawing of K_{13} to show that the crossing number of K_{13} is one of the numbers in {217,219,221,223,225}. We show that the crossing number is not 217.