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

Ayumu Inoue, Naoki Kimura, Ryo Nikkuni, Kouki Taniyama*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Article number2250076
JournalJournal of Knot Theory and its Ramifications
DOIs
Publication statusAccepted/In press - 2022

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

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