Abstract

By a diagram we mean a topological space obtained by gluing to a standard circle a finite number of pairwise non-intersecting closed rectangles along their lateral sides, the glued rectangles being pairwise disjoint. Diagrams are not new objects; they have been used in many areas of low-dimensional topology. Our main goal is to develop the theory of diagrams to a level sufficient for application in yet another branch: the theory of tangles. We provide diagrams with simple additional structures: the smoothness of the circles and rectangles that are pairwise consistent with each other, the orientation of the circle, and a point on the circle. We introduce a new equivalence relation (as far as the author knows, not previously encountered in the scientific literature): kindred relation. We define a surjective mapping of the set of classes of kindred diagrams onto the set of classes of diffeomorphic smooth compact connected two-dimensional manifolds with a boundary and note that in the simplest cases this surjection is also a bijection. The application of the constructed theory to the tangle theory requires additional preparation and therefore is not included in this article; the author intends to devote a separate publication to this application.

中文翻译：

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亲属图

摘要

我们所说的图是指通过将有限数量的成对不相交闭合矩形沿其侧面粘合到标准圆上而获得的拓扑空间，粘合的矩形是成对不相交的。图表并不是新事物；而是新事物。它们已被用于低维拓扑的许多领域。我们的主要目标是将图论发展到足以应用于另一个分支：缠结理论的水平。我们提供具有简单附加结构的图表：成对一致的圆和矩形的平滑度、圆的方向以及圆上的点。我们引入一种新的等价关系（据作者所知，以前在科学文献中没有遇到过）：亲属关系。我们定义了类图类集到具有边界的微分同胚光滑紧连通二维流形类集的满射映射，并注意到在最简单的情况下，该满射也是双射。将构造理论应用于缠结理论需要额外的准备工作，因此不包含在本文中；作者打算针对该申请专门发表一篇文章。