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 [Formula: see text]-cycles is odd. The sum of the rotation numbers of all [Formula: see text]-cycles is even. We show analogous results for [Formula: see text]-cycles, [Formula: see text]-cycles and [Formula: see text]-cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a [Formula: see text]-cycle that is not an unknot with maximal Thurston–Bennequin number, and the sum of all Thurston–Bennequin numbers of the cycles is seven times the sum of all Thurston–Bennequin numbers of the [Formula: see text]-cycles. We show analogous results for a Heawood graph. We also show some other results for some graphs. We characterize abstract graphs that have a generic immersion into a plane whose all cycles have rotation number [Formula: see text].