The Jones polynomial of collections of open curves in 3-space
The Jones polynomial of collections of open curves in 3-space
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3 空间中开曲线集合的琼斯多项式
DOI：
10.1098/rspa.2022.0302
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Panagiotou, Eleni
中科院分区：
文献类型：
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作者：
Barkataki, Kasturi;Panagiotou, Eleni
测量三维空间中开放曲线集合的纠缠复杂性一直是一个棘手但紧迫的数学问题，与大量物理系统有关，例如在聚合物和生物聚合物中。在这篇手稿中，我们给出了琼斯多项式的一个新的定义，它将经典的琼斯多项式推广到三维空间中的开曲线集合。更确切地说，我们首先给出了链状图的Jones多项式的一个新的定义，并证明了这是一个定义良好的单变量多项式，它是一个拓扑不变量，对于链型链状图，它与相应的链状图的拓扑不变量重合。使用(Panagiotou E，Kauffman L.2020Proc.)中介绍的框架。R.Soc.A476,20200124。(DOI：10.1098/rspa.2020.0124))，这使我们能够定义三维空间中开曲线和闭曲线的集合的琼斯多项式。对于三维空间中的开放曲线集合，琼斯多项式具有实系数，并且它是曲线坐标的连续函数。当曲线的端点趋于重合时，琼斯多项式趋向于结合链的端点。我们用数值例子证明了这种新的Jones多项式首次能够刻画三维空间中开曲线集合的拓扑/几何复杂性。
Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript, we give a novel definition of the Jones polynomial that generalizes the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, coincides with that of the corresponding link. Using the framework introduced in (Panagiotou E, Kauffman L. 2020Proc. R. Soc. A476, 20200124. ((doi:10.1098/rspa.2020.0124)), this enables us to define the Jones polynomial of collections of open and closed curves in 3-space. For collections of open curves in 3-space, the Jones polynomial has real coefficients and it is a continuous function of the curves’ coordinates. As the endpoints of the curves tend to coincide, the Jones polynomial tends to that of the resultant link. We demonstrate with numerical examples that the novel Jones polynomial enables us to characterize the topological/geometrical complexity of collections of open curves in 3-space for the first time.