Crossing numbers and rotation numbers of cycles in a plane immersed graph

Abstract

For any generic immersion of a Petersen graph into a plane, the number of crossing points between two edges of distance one is odd. The sum of the crossing numbers of all 5-cycles is odd. The sum of the rotation numbers of all 5-cycles is even. We show analogous results for 6-cycles, 8-cycles and 9cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a 5-cycle that is not an unknot with maximal Thurston-Bennequin number, and the sum of all Thurston-Bennequin numbers of the cycles is 7 times the sum of all Thurston-Bennequin numbers of the 5-cycles. We show analogous results for a Heawood graph. We also show some other results for some graphs. We characterize abstract graphs that has a generic immersion into a plane whose all cycles have rotation number 0.