论文标题
汤普森正链接的连接组件数量的计算研究
A computational study of the number of connected components of positive Thompson links
论文作者
论文摘要
大约十年前，沃恩·琼斯（Vaughan Jones）引入了一种方法，可以从汤普森集团（Thompson Groups $ f $）的元素中产生结，后来扩展到了布朗 - 汤普森集团（Brown-Thompson Group）$ f_3 $。在本文中，我们定义了一种从$ f $和$ f_3 $的元素中产生排列的方法，我们称为汤普森排列。每个汤普森置换的轨道数量与链接的连接组件的数量一致。我们探索固定\ emph {width}和\ emph {height}的$ f_3 $的正元素，并根据数值实验做出一些猜想。为了定义汤普森排列，我们需要为从$ f $和$ f_3 $的元素产生的每个链接分配一个方向。我们证明可以以这种方式产生所有面向的链接。
Almost a decade ago Vaughan Jones introduced a method to produce knots from elements of the Thompson groups $F$, which was later extended to the Brown-Thompson group $F_3$. In this article we define a way to produce permutations out of elements of the $F$ and $F_3$ that we call Thompson permutations. The number of orbits of each Thompson permutation coincides with the number of connected components of the link. We explore the positive elements of $F_3$ of fixed \emph{width} and \emph{height} and make some conjectures based on numerical experiments. In order to define the Thompson permutations we need to assign an orientation to each link produced from elements of $F$ and $F_3$. We prove that all oriented links can be produced in this way.
