Invariants of degree one of almost generic plane immersed curves
Let \(\Omega\) be the infinite dimensional space of all immersions of a circle into the plane, \(\Sigma\) its discriminant (stratified) hypersurface, consisting of non-generic immersions, and \(\Sigma^1\) the codimension 1 strata of \(\Sigma\), consisting of immersions with only one singularity of degree 1. In [Adv. Sov. Math. 21, 33-91 (1994; Zbl 0864.57027)] \textit{V. I. Arnold} makes use of the properties of \(\Sigma^1\) to introduce the three basic (integer) invariants \(St,J^+,J^-\) of plane generic immersed curves (having only ordinary double points of transversal intersections). In analogy, the present paper introduces seven new invariants of degree 1 of curves in \(\Sigma^1\), by making use of the properties of the codimension 2 strata \(\Sigma^2\) (consisting of immersions with only one singularity of degree 2). Moreover, the author provides an axiomatic description of these invariants, which are dependent on the orientation of the immersed circle and take value either in \(\mathbb{Z}\) (five of them) or in \(\mathbb{Z}_3\) (two of them).
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