Classifying Immersed Curves

Abstract

Let a collection $\gamma$ of generically immersed curves be given in an oriented surface $G$. 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 ${\gamma _1}$ and ${\gamma _2}$ fill the surface $G$ in the sense that the complementary regions are disks, then there is a homeomorphism of $G$ taking one to the other if and only if ${\gamma _1}$ and ${\gamma _2}$ have isomorphic Gauss paragraphs. This notion of isomorphism is defined here; it ignores the choices made in defining the Gauss words.