Virtual Strings for Closed Curves with Multiple Components

William J. Schellhorn

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Virtual Strings for Closed Curves with Multiple Components

William J. Schellhorn

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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.

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BibTeX RIS

William J. Schellhorn. 2005-03-08. Virtual Strings for Closed Curves with Multiple Components. https://arxiv.org/abs/math/0412023

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