The Arf—invariant and the Arnold invariants of plane curves

Abstract

In [A] V.I.Arnold considered closed generic plane curves, i.e. immersions S 1 → R 2, the images of which have no singularities except simple (double) self-intersections. In a generic one-parameter family of immersions three types of modifications (“perestroykas”) of generic curves can be met. They correspond to three natural strata in the set of non-generic immersions (the discriminant). These strata consist of immersions with a direct self-tangency (J +), with an inverse self-tangency (J −), and with a triple crossing (St) respectively. All three of them are coorientable. An invariant of generic plane curves is an invariant of the first order (in the sense of Vassiliev) if its change under a modification of crossing a stratum (in the positive direction) depends only on the stratum, but not on a (non-singular) point of its crossing. For closed plane curves V.I.Arnold defined basic invariants of the first order J+, J−, and St corresponding to the described strata. The invariants J+, J−, and St can be defined for so-called “long” curves (that is for curves going from the infinity to the infinity) as well.