An integral generalization of the Gusein-Zade–Natanzon theorem

One of the useful methods of the singularity theory, the method of real morsifications [AC0, GZ] (see also [AGV]) reduces the study of discrete topological invariants of a critical point of a holomorphic function in two variables to the study of some real plane curves immersed into a disk with only simple double points of self-intersection. For the closed real immersed plane curves V.I.Arnold [Ar] found three simplest first order invariants J and St. Arnold’s theory can be easily adapted to the curves immersed into a disk. In [GZN] S.M.Gusein-Zade and S.M.Natanzon proved that the Arf invariant of a singularity is equal to J/2(mod 2) of the corresponding immersed curve. They used the definition of the Arf invariant in terms of the Milnor lattice and the intersection form of the singularity. However it is equal to the Arf invariant of the link of singularity, the intersection of the singular complex curve with a small sphere in C centered at the singular point. In knot theory it is well known (see, for example [Ka]) that Arf invariant is the mod 2 reduction of some integer-valued invariant, the second coefficient of the Conway polynomial, or the Casson invariant [PV]. Arnold’s J/2 is also an integer-valued invariant. So one might expect a relation between these integral invariants. A few years ago N.A’Campo [AC1, AC2] 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. But 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. It turns out that this J 2 invariant is a second order invariant of the mixed Jand J-types. To the best of my knowledge, so far nobody tried to study the mixed J-type invariants. It seems that our invariant is one of the simplest such invariants. The problem of describing all second order J-type invariants is open. In section 1 we review the A’Campo construction and list the properties of the links obtained. In sections 2 and 3 we introduce the Casson invariant and Arnold’s type invariants of curves immersed into a disk. In section 4 we formulate our main result. The proof is based on M.Hirasawa’s construction [Hir] of a Seifert surface for the A’Campo links. We review this construction in section 5.