Virtual Strings for Closed Curves with Multiple Components

Abstract

A Gauss paragraph is a combinatorial formulation of a generic closed curve with multiple components on some surface. A virtual string is a collection of circles with arrows that represent the crossings of such a curve. Every closed curve has an underlying virtual string and every virtual string has an underlying Gauss paragraph. A word-wise partition is a partition of the alphabet set of a Gauss paragraph that satisfies certain conditions with respect to the Gauss paragraph. In this paper we use the theory of virtual strings to obtain a combinatorial description of closed curves in the 2-sphere (and therefore 2-dimensional Euclidean space) in terms of Gauss paragraphs and word-wise partitions.