论文标题
打结投影的子公司图
Sub-chord diagrams of knot projections
论文作者
论文摘要
和弦图是一个带有配对点的圆，每对由和弦连接。每条通用的沉浸式球形曲线都通过将每个和弦与两个双点的预图相关联，提供了和弦图。任何两个球形曲线都可以通过三种类型的局部替换RI，RII和RIII的有限序列（称为reidemeister移动）相关。这项研究通过应用与与局部替代物相关的分支相关的连接所定义的，通过应用RI，强RII，强RIII和弱的RIII来计算在任何球形曲线中嵌入的子弦图数量的差异。这在RI和强RIII下产生了一个新的整数价值不变，该整数不变，可完整地分类的质量降低球形曲线，最多可至少七个双点（定理2，图24）：以前没有这样的不变。不变式表达了必要和充分的条件，即球形曲线可以通过有限的RI和强RIII移动的简单闭合曲线有关（定理3）。此外，通过计数子系统图（定理4）提供了蝇台下的球形曲线的不变性。
A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve provides a chord diagram by associating each chord with two preimages of a double point. Any two spherical curves can be related by a finite sequence of three types of local replacement RI, RII, and RIII, called Reidemeister moves. This study counts the difference in the numbers of sub-chord diagrams embedded in a full chord diagram of any spherical curve by applying one of the moves RI, strong RII, weak RII, strong RIII, and weak RIII defined by connections of branches related to the local replacements (Theorem 1). This yields a new integer-valued invariant under RI and strong RIII that provides a complete classification of prime reduced spherical curves with up to at least seven double points (Theorem 2, Fig. 24): there has been no such invariant before. The invariant expresses the necessary and sufficient condition that spherical curves can be related to a simple closed curve by a finite sequence of RI and strong RIII moves (Theorem 3). Moreover, invariants of spherical curves under flypes are provided by counting sub-chord diagrams (Theorem 4).
