On the Bennequin invariant and the geometry of wave fronts
A smooth closed curve in an oriented surface \(M\) is an oriented immersed circle in \(M\). The connected components of the set of all smooth closed curves \({\mathcal C}\) are classified by the degree of the tangential Gauss map. \textit{V. I. Arnol'd} [`Topological invariants of plane curves and caustics' (Univ. Lect. Ser. 5, AMS, Providence/RI) (1994; Zbl 0858.57001); Adv. Sov. Math. 21, 33-91 (1994; Zbl 0864.57027)] introduced the three basic invariants \((J^\pm,St)\) of the three components of the discriminant, which correspond to one triple point and two different self-tangencies of immersions. The author applies this theory to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant \(J^+\) in terms of the geometry of the front. In the case of affine wave fronts, some formulae previously announced by S. L. Tabachnikov are proved. Results of O. Viro, F. Fabricius-Bjerre, and J. L. Weiner for smooth plane and spherical curves are generalized to wave fronts.
scientific article; zbMATH DE number 1019780
scientific article