Classifying immersed curves

Abstract

Let a collection γ \gamma of generically immersed curves be given in an oriented surface G G . To each component circle, associate a Gauss word by traveling once around the circle and recording the crossing points with signs. The set of these words forms a Gauss paragraph. If γ 1 {\gamma _1} and γ 2 {\gamma _2} fill the surface G G in the sense that the complementary regions are disks, then there is a homeomorphism of G G taking one to the other if and only if γ 1 {\gamma _1} and γ 2 {\gamma _2} have isomorphic Gauss paragraphs. This notion of isomorphism is defined here; it ignores the choices made in defining the Gauss words.