An integral generalization of the Gusein-Zade–Natanzon theorem

Several years ago N. A'Campo invented a construction of a link from a real curve immersed into the disk. In the case of the curve obtained by the real morsification method of singularity theory the link is isotopic to the link of the corresponding singularity. S. M. Gusein-Zade and S. M. Natanzon proved that the Arf invariant of the obtained knot equals $J^-/2$ (mod 2) of the corresponding curve. Here we describe the Casson invariant of A'Campo knots as a $J^\pm$-type invariant of the immersed curve. Thus we get an integral generalization of the Gusein-Zade–Natanzon theorem. It turns out that this $J_2^\pm$-invariant is a second order invariant of the mixed $J^+$- and $J^-$-types. To the best of my knowledge, nobody has yet tried to study the mixed $J^\pm$-type invariants. It seems that our invariant is one of the simplest such invariants.