Invariants of degree one of almost generic plane immersed curves

In 1993,V.I. Arnold used the approach of singularity theory to construct in variants of plane generic immersed curves. This approach suggests a hierarc hy of invariants, the coarsest and most fundamental being Arnold's invarian ts of degree 1. Consider the infinite dimensional space Omega, of all immer sions of S-1 --> R-2. The non-generic immersions form a hypersurface called the discriminant which is stratified. The immersions with only one singula rity of degree 1 form Sigma (1), the main part of the discriminant. Given a generic curve on Sigma (1) (the codim 1 strata), we introduce new invarian ts of degree 1 in the following sense: when this generic curve passes throu gh Sigma (2) (immersions with one singularity of degree 2), the value of th e invariant jumps by a number which depends only on the stratum of codim 2. The natural stratification of the discriminant yields information about th e topology of Sigma (1), necessary to prove that the invariants are well de fined. Of the seven invariants found, five have values in Z and two have va lues in Z(3). This paper provides an axiomatic description of these invaria nts.