Moscow--Petersburg seminar on low-dimensional mathematics
SPb, PDMI, 21.06.2002, 16:00--17:45, room 311

"Knots and plane immersed curves"
Sergei Chmutov

One of the useful methods of the singularity theory, the method of
real morsifications reduces
the study of discrete topological invariants of a critical point
of a holomorphic function of two variables to the study of some
real plane curves immersed into a disk with only simple double
points of self-intersections. For the closed real immersed plane
curves V.I.Arnold found three simplest first order
invariants $J^\pm$ and $St$. Arnold's theory can be easily
adapted to the curves immersed into a disk.
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.

We prove an integral generalization of the
Gusein-Zade--Natanzon theorem, expressing the Casson invariant in
terms of $J^\pm$-type invariants of the immersed plane curve.
It turns out that this $J^\pm_2$ invariant is a second order invariant 
of the mixed $J^+$- and $J^-$-types. The 
problem of describing of all second order $J^\pm$-type invariants is 
still open.

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
a 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. We
describe the Casson invariant of A'Campo's knots as an invariant
of the immersed curves.