IUMJ

Title: A new aspect of the Arnold invariant J+ from a global viewpoint

Authors: Kenta Hayano and Noboru Ito

Issue: Volume 64 (2015), Issue 5, 1343-1357

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 \emph{direct self-tangency perestroika} in a generic regular homotopy from a standard curve to a given one; the other basic local moves, namely \emph{inverse self-tangency perestroika} and \emph{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.