### 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 |
---|---|

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 |

### Fingerprint

### 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

**A new aspect of the arnold invariant J ^{+} from a global viewpoint.** / Hayano, Kenta; Ito, Noboru.

Research output: Contribution to journal › Article

^{+}from a global viewpoint',

*Indiana University Mathematics Journal*, vol. 64, no. 5, pp. 1343-1357. https://doi.org/10.1512/iumj.2015.64.5641

}

TY - JOUR

T1 - A new aspect of the arnold invariant J+ from a global viewpoint

AU - Hayano, Kenta

AU - Ito, Noboru

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Legendrian knots

KW - Plane curves

KW - The Arnold invariants

UR - http://www.scopus.com/inward/record.url?scp=84956686450&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84956686450&partnerID=8YFLogxK

U2 - 10.1512/iumj.2015.64.5641

DO - 10.1512/iumj.2015.64.5641

M3 - Article

AN - SCOPUS:84956686450

VL - 64

SP - 1343

EP - 1357

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 5

ER -