A New Aspect of the Arnold Invariant J+from a Global Viewpoint

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, namel...

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Bibliographic Details
Published in:Indiana University mathematics journal 2015-01, Vol.64 (5), p.1343-1357
Main Authors: Hayano, Kenta, Ito, Noboru
Format: Article
Language:English
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Summary: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.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2015.64.5641