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

Ayumu Inoue; Naoki Kimura; Ryo Nikkuni; Kouki Taniyama

doi:10.1142/s0218216522500766

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arXiv · 2205.01013

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

Ayumu Inoue·Naoki Kimura·Ryo Nikkuni·Kouki Taniyama

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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 $0$.

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Ayumu Inoue, Naoki Kimura, Ryo Nikkuni, Kouki Taniyama. 2022-05-02. Crossing numbers and rotation numbers of cycles in a plane immersed graph. https://doi.org/10.1142/s0218216522500766

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