626
Communications of the Moscow Mathematical Society
On the number of topological types of plane curves
S. M. Gusein-Zade and F. S. Duzhin
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 S 1 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 : S 1 → S 1 and Σ : R2 → R2 be standard maps changing the
orientation of the circle and plane. The curves may be supersymmetric: [l] = [lσ] = [Σl] = [Σlσ],
This work was supported by the Russian Foundation for Fundamental Research (grant no. 9801-00612).
Communications of the Moscow Mathematical Society
627
(1)-symmetric: [l] = [Σlσ], (2)-symmetric: [l] = [lσ], (3)-symmetric: [l] = [Σl], and asymmetric
([l] is the class of oriented curves on the oriented plane topologically equivalent to the curve l). We
introduce these curves, considered as non-oriented on the non-oriented plane, in the four columns
of the table below corresponding to the closed curves (1, 1, 1, 1), (2, 1, 1, 1), (2, 2, 1, 1), (2, 1, 2, 1)
and (4, 2, 2, 1). In order to clarify whether, for example, a given curve is (3)-symmetric we need to
reflect the diagram of the curve relative to the vertical axis (thereby obtaining, generally speaking,
a non-normal diagram), normalize it, and verify if the diagram obtained is found among the normal
diagrams corresponding to different choices of the point on the external contour of the curve under
consideration.
A program written in Pascal determining the number of closed curves of different types is
found at the URL address: http://www.botik.ru/˜duzhin/dataprog/allcurve.pas
The results of the calculations are entered in the following table.
Closed curves
Long curves
Oriented
R2 , S 1
R2
S1
–
0
2
1
1
1
1
1
3
2
2
2
2
2
10
5
5
5
8
3
39
21
21
20
42
4
204
102
102
82
260
5
1262
639
640
435
1796
6
8984
4492
4492
2645
13396
7
67959
34032
34047
18489
105706
8
544504
272252
272252
141326
870772
9
4535030
2267905
2268085
1153052
7420836
10
39004772
19502386
19502386
9819315
65004584
Bibliography
[1] V. I. Arnol’d, Topological invariants of plane curves and caustics, Amer. Math. Soc.,
Providence, RI 1994.
[2] S. M. Gusein-Zade, Adv. Soviet Math. 21 (1994), 189–198.
Moscow State University
Received 7/APR/98