Panagiotou Eleni, Kauffman Louis H
Department of Mathematics and SimCenter, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA.
Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago, Chicago, IL 60607-7045, USA.
Proc Math Phys Eng Sci. 2020 Aug;476(2240):20200124. doi: 10.1098/rspa.2020.0124. Epub 2020 Aug 5.
In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.
在本手稿中，我们介绍了一种测量三维空间中曲线纠缠度的方法，该方法将纽结和链环多项式的概念扩展到开放曲线。我们定义了三维空间中曲线的括号多项式，并证明它具有实系数且是曲线坐标的连续函数。这用于以适用于三维空间中开放曲线和封闭曲线的方式定义琼斯多项式。对于开放曲线，琼斯多项式具有实系数，并且是曲线坐标的连续函数，当曲线的端点趋于重合时，开放曲线的琼斯多项式趋于所得纽结的琼斯多项式。对于封闭曲线，它是一种拓扑不变量，如同经典的琼斯多项式。我们展示了这些度量如何对于多边形曲线获得更简单的表达式，并在三边和四边多边形曲线的情况下为其计算提供有限形式。
