### Abstract

In this paper, we study the Arnold invariant J^{+} for plane and spherical curves. This invariant essentially counts the number of a certain type of local moves called direct self-tangency perestroika in a generic regular homotopy from a standard curve to a given one; the other basic local moves, namely inverse self- tangency perestroika and triple point crossing, do not change the value of J^{+}. Thus, behavior of J^{+} under local moves is rather obvious. However, it is less understood how J^{+} behaves in the space of curves on a global scale. We study this problem using Legendrian knots, and give infinitely many regular homotopic curves with the same J^{+} that cannot be mutually related by inverse self-tangency perestroika and triple point crossing.

Original language | English |
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Pages (from-to) | 1343-1357 |

Number of pages | 15 |

Journal | Indiana University Mathematics Journal |

Volume | 64 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

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### Keywords

- Legendrian knots
- Plane curves
- The Arnold invariants

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{+}from a global viewpoint.

*Indiana University Mathematics Journal*,

*64*(5), 1343-1357. https://doi.org/10.1512/iumj.2015.64.5641