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

TL;DR

This paper investigates crossing and rotation numbers of cycles in plane immersions of Petersen and Heawood graphs, revealing parity properties and relationships with Legendrian embeddings, and characterizes graphs with all cycles having zero rotation number.

Contribution

It extends known parity and rotation number results to larger cycles and other graphs, and characterizes graphs with all cycles of zero rotation number.

Findings

Number of crossings between edges of distance one is odd in Petersen graph immersions.

Sum of rotation numbers of all 5-cycles is even in Petersen graph immersions.

Existence of a 5-cycle not an unknot with maximal Thurston-Bennequin number in Legendrian embeddings.

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 $9$-cycles. 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 $0$.