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, 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.