## 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 seven 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 have a generic immersion into a plane whose all cycles have rotation number 0.

Original language | English |
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Article number | 2250076 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 31 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2022 Oct 1 |

## Keywords

- Crossing number
- Heawood graph
- Legendrian knot
- Legendrian spatial graph
- Petersen graph
- Thurston-Bennequin number
- plane immersed graph
- reduced Wu and generalized Simon invariant
- rotation number

## ASJC Scopus subject areas

- Algebra and Number Theory