Arnold's seminar
Moscow Univ, 29.10.2002

Francesca Aicardi.
Invariants of apparent countours.
(joint with T.Ohmoto).

The apparent contour  is a collection of closed plane curves
with cusps. It is defined as the image in the plane of the
critical set (the set of point where rank(dF)<2) of a smooth map
F from a  2-dimensional compact manifold M  to the plane.
We calculate  the order one Vassiliev invariants of the apparent contour
in the general case  and in the special cases when M is oriented,
and when F is a projection to the plane and M is  immersed in R^3.
We give a geometrical interpretation (and hence integral
formulae) for these invariants.