On the number of topological types of plane curves

Abstract

Plane curves of general position have as singularities only simple double self-intersections. The problem of describing closed curves on a plane and in space (of knots) to within a homeomorphism of the plane or space goes back apparently to Gauss. Both problems are rather complex and remain unsolved. Arnol’d ([1], p. 5) says that “the combinatorics of plane curves seems to be far more complicated than that of knot theory (which might be considered as a simplified ‘commutative’ version of the combinatorics of plane curves and which is probably embedded in plane curves theory)”. Statistics on the number of plane curves with not more than five points of self-intersection are given in [1]. The enumeration of the curves was carried out manually and not based on any general algorithms. The problem of classifying ‘long curves’ on the plane was considered in [2]. A long curve is the image of a general non-degenerate map of the straight line R into the plane R2 coinciding with the standard embedding x 7→ (x, 0) exterior to a certain bounded set in R. An algorithm was described for enumerating these curves. It is based on a description of long curves with the help of diagrams of a special form, on a selection of a subclass of (so-called normal) diagrams whose elements are in one-to-one correspondence with classes of topologically equivalent long curves, and on an algorithm for enumerating all normal diagrams (without repetitions or testing of improper variants.) This algorithm was the basis for a computer program determining the number of long curves with various numbers of points of self-intersection (see the last column of the table at the end of this note). In this paper we give a brief description of a modification to this algorithm for calculating the number of closed curves on the plane. A closed plane curve (referred to in what follows simply as a curve) is the image of a smooth non-degenerate map of the circle S1 into the plane R2 of general position, that is, having as singularities only simple double self-intersections. Two curves are equivalent if there exists a homeomorphism of the plane transforming one of them into the other. Four types of equivalence arise depending on whether or not homeomorphisms changing the orientation of the plane or the circle are admitted. Thus, if only homeomorphisms preserving the orientation of the plane are admitted, we can speak of classifying curves on the oriented plane. Corresponding to a long curve is a closed curve obtained by joining its ends in the upper half-plane. (One closed curve can correspond to various long curves.) Long curves are thus in one-to-one correspondence with classes of topologically equivalent non-oriented curves on an oriented plane with a marked point on one of the segments of the external contour. We assume that the closed curve corresponding to the given long curve is obtained from m long curves. The number m is equal to the number of topologically non-equivalent methods for marking the point on the external contour of the closed curve. The number of non-oriented curves on the oriented plane is equal to the sum over all the long curves of the quantities 1/m. The value of m for a given long curve can be determined in the following way. The corresponding point on the external contour of the closed curve is sequentially displaced by one segment anti-clockwise. It is not difficult to describe a certain diagram of the corresponding long curve. The procedure for normalizing the diagram obtained is the central part of the algorithm. If the normalized diagram coincides with the original one, then these long curves are equivalent. A long curve equivalent to the original one is obtained after the chosen point is displaced by m segments. To calculate the number of long curves relative to other types of equivalence (distinguished by the orientation or non-orientation of the plane or circle) we need to determine the type of symmetry of the curve (see [1]). Let σ : S1 → S1 and Σ: R2 → R2 be standard maps changing the orientation of the circle and plane. The curves may be supersymmetric: [l] = [lσ] = [Σl] = [Σlσ],