Abstract A textile structure is a periodic arrangement of threads in the thickened plane. A topological classification of textile structures is harder than for classical knots and links that are non-periodic and restricted to a bounded region. The first important problem is to encode all textile structures in a simple combinatorial way. This paper extends the notion of the Gauss code in classical knot theory, providing a tool for topological computation on these structures. As a first application, we present a linear time algorithm for determining whether a code represents a textile in the physical sense. This algorithm, along with invariants of textile structures, allowed us for the first time to classify all oriented textile structures woven from a single component up to complexity five.

中文翻译：

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纺织结构的编码和拓扑计算

摘要 纺织结构是线在加厚平面上的周期性排列。纺织品结构的拓扑分类比非周期性且仅限于有界区域的经典结和链接更难。第一个重要问题是以简单的组合方式对所有纺织品结构进行编码。本文扩展了经典结理论中高斯码的概念，为这些结构的拓扑计算提供了一种工具。作为第一个应用，我们提出了一种线性时间算法，用于确定代码是否代表物理意义上的纺织品。该算法与纺织品结构的不变量一起，使我们第一次能够对从单一组件编织到复杂度 5 的所有定向纺织品结构进行分类。