A Classification of Spherical Curves Based on Gauss Diagrams

Abstract

We consider generic smooth closed curves on the sphere $$S^2$$ . These curves (oriented or not) are classified relatively to the group $$\text{ Diff }(S^2)$$ or its subgroup $$\text{ Diff }^+(S^2)$$ ), with the Gauss diagrams as main tool. V. I. Arnold determined the numbers of orbits of curves with n double points when $$n<6$$ . This paper explains how a preliminary classification of the Gauss diagrams of order 5, 6 and 7 allows to draw up the list of the realizable chord diagrams of these orders. For each such diagram $$\Gamma $$ and for each Arnold symmetry type T, we determine the number of orbits of spherical curves of type T realizing $$\Gamma $$ . As a consequence, we obtain the total numbers of curves (oriented or not) with 6 or 7 double points on the sphere (oriented or not) and also the number of curves with special properties (e.g. having no simple loop).