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Rectangular diagrams of surfaces: distinguishing Legendrian knots
Journal of Topology ( IF 1.1 ) Pub Date : 2021-07-02 , DOI: 10.1112/topo.12194
Ivan Dynnikov 1 , Maxim Prasolov 1
Affiliation  

In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce basic moves that allow the transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams and various moves for them that can be used to transform presentations of isotopic surfaces to each other. The moves are divided into two (non-exclusive) types so that, vaguely speaking, type I moves commute with type II ones. This commutation is the matter of the main technical result of the paper. We use it as well as a relation of the moves to Giroux's convex surfaces to propose a new method for distinguishing Legendrian knots. We apply this method to show that two Legendrian knots having topological type 6 2 are not equivalent. More applications of the method will be the subject of subsequent papers. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

中文翻译:

曲面的矩形图:区分勒让德结

在较早的一篇论文中,我们介绍了表面的矩形图,并表明三球体中表面的任何同位素类都可以用矩形图表示。在这里,我们研究这些图的转换并介绍允许在表示同位素表面的图之间进行转换的基本移动。我们还介绍了称为镜像图的更通用的组合对象以及它们的各种移动,可用于将同位素表面的表示相互转换。移动分为两种(非排他性)类型,因此,模糊地说,I 型移动与 II 型移动通勤。这种换算是论文的主要技术成果的问题。我们使用它以及移动与 Giroux 凸面的关系来提出一种区分勒让德结的新方法。 6 2 不等价。该方法的更多应用将是后续论文的主题。这篇论文广泛依赖于彩色数字。印刷版本中对颜色的某些引用可能没有意义,我们建议读者参考包含颜色数字的在线版本。
更新日期:2021-07-04
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