论文标题
琼斯在3个空间中的开放曲线集合多项式
The Jones polynomial of collections of open curves in 3-space
论文作者
论文摘要
在3空间中测量开放曲线集合的纠缠复杂性是一个棘手但又紧迫的数学问题，与大量物理系统有关，例如聚合物和生物聚合物。在本手稿中，我们给出了琼斯多项式的新颖定义，该定义将经典的琼斯多项式概括为3个空间的开放曲线的集合。更准确地说，首先，我们提供了琼斯多项式（开放链路图）的新颖定义，并表明这是一个定义明确的单个变量多项式，它是拓扑不变的，对于链接类型的linkoids，它与相应的链接相吻合。使用Panagiotou和Kauffman 2020 ARXIV中引入的框架：2001.01303，这使我们能够定义3个空间开放和封闭曲线的琼斯多项式。对于以3空间为单位的开放曲线的集合，琼斯多项式具有实际系数，并且是曲线坐标的连续函数。随着曲线的终点倾向于重合，琼斯多项式倾向于所得链接的端点。我们用数值示例证明了新颖的琼斯多项式使我们能够首次以3个空间来表征开放曲线集合的拓扑和几何复杂性。
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, it coincides with that of the corresponding link. Using the framework introduced in Panagiotou and Kauffman 2020 arXiv:2001.01303, 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 curve 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 and geometrical complexity of collections of open curves in 3-space for the first time.
