Virtual Strings for Closed Curves with Multiple Components

摘要

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