Invariants of curves and fronts via Gauss diagrams

Abstract

We use a notion of chord diagrams to define their representations in Gauss diagrams of plane curves. This enables us to obtain invariants of generic plane and spherical curves in a systematic way via Gauss diagrams. We define a notion of invariants are of finite degree and prove that any Gauss diagram invariants are of finite degree. In this way we obtain elementary combinatorial formulas for the degree 1 invariants J ± and St of generic plane curves introduced by Arnold [1] and for the similar invariants J ± S and St S of spherical curves. These formulas allow a systematic study and an easy computation of the invariants and enable one to answer several questions stated by Arnold. By a minor modification of this technique we obtain similar expressions for the generalization of the invariants J ± and St to the case of Legendrian fronts. Different generalizations of the invariants and their relations to Vassiliev knot invariants are discussed.