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arXiv math/04120232004-12math.GT

多分量闭曲线的虚弦

Virtual Strings for Closed Curves with Multiple Components

William J. Schellhorn

备注：: 35 pages, 5 figures. No fundamental changes were made between the second and third versions. The third version contains some clarifications in the definitions and proofs that will hopefully improve understanding

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摘要：

高斯段是在某个曲面上具有多个分量的一般闭曲线的组合公式。虚拟字符串是带有箭头的圆的集合，箭头表示此类曲线的交叉点。每个闭曲线都有一个潜在的虚拟字符串，每个虚拟字符串都有一个潜在的高斯段落。按单词划分是高斯段落的字母集的划分，其满足关于高斯段落的某些条件。在本文中，我们使用的理论虚拟字符串获得的组合描述的闭曲线在2球（因此2维欧几里得空间）的高斯段落和字明智的分区。

英文摘要：

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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