New Whitney-Type Formulae for Plane Curves

The classical Whitney formula relates the algebraic number of self-intersections of a generic plane curve to its winding number. We generalize it to an innnite family of identities, expressing the winding number in terms of the internal geometry of a plane curve. This enables us to split the Whit-ney formula by some characteristic of double points. It turns out, that only crossings of a very speciic type contribute to the computation of the winding number. We also provide a "diierence integration" of these formulae, establishing a new family of simple formulae with the base point pushed oo the curve. Similar new identities are obtained for Arnold's invariant Strangeness of plane curves. 1. Whitney formula and its generalizations 1.1. Introduction. The classical Whitney formula 4] relates the algebraic number of times that a generic immersed plane curve intersects itself to the Whit-ney index, or winding number, of this curve. Since it was discovered in 1937, this formula remained more of an isolated curious fact, than a part of some more general picture. In particular, its generalizations remained unknown. The rst step in this direction was made only recently 2], with an introduction of higher-dimensional versions of the Whitney formula. Here we take a diierent direction of its generalization , showing that this is just a simplest one in an innnite family of identities. These identities express Whitney index of a plane curve in terms of some functions, deened in double points of the curve. A particular choice of these functions as elementary bump ones allows us to split the sum in the Whitney formula over diierent types of double points (see Section 1.4). It turns out, that only a very speciic type of double points contributes to the computation of index. All these formulae, just as the original one, involve a choice of a base point on the curve. We introduce another type of formulae, this time with the base point pushed oo the curve. Taking the diierence of two such formulae with the base point placed in a pair of adjacent regions, one gets a formula of the rst type, with the base point on the arc of the curve between these regions. Thus the second type of formulae can be considered as a diierence integration of formulae of the rst type with respect to position of the base point (see Section 2). Finally, we turn to Arnold's invariant St …