An integral generalization of the Gusein-Zade--Natanzon theorem

A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this article we describe the Casson invariant of A'Campo's knots as a J^{+/-}-type invariant of the immersed curves. Thus we get an integral generalization of the Gusein-Zade--Natanzon theorem which says that the Arf invariant of a singularity is equal to J^{-}/2(mod 2) of the corresponding immersed curve. It turns out that this invariant is a second order invariant of the mixed J^{+}- and J^{-}-types.