Vassiliev measures of complexity of open and closed curves in 3-space
Vassiliev measures of complexity of open and closed curves in 3-space
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DOI：
10.1098/rspa.2021.0440
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发表时间：
2021-04
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影响因子：
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通讯作者：
E. Panagiotou;L. Kauffman
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文献类型：
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作者：
E. Panagiotou;L. Kauffman
在本文中，我们定义了 3 空间中开曲线的复杂性 Vassiliev 测度。这些与 3 空间中开曲线的增强琼斯多项式的系数有关。这些 Vassiliev 测量是曲线坐标的连续函数；由于曲线的两端趋于重合，它们会收敛到所得结的相应瓦西里耶夫不变量。我们重点关注 3 空间中闭曲线和开曲线的增强型琼斯多项式的第二个 Vassiliev 测度。对于闭合曲线，第二个 Vassiliev 测度可以通过高斯代码图计算，并且它具有积分公式，即双交替自链接积分。双交替自联积分是闭曲线的拓扑不变量，也是 3 空间中开曲线的曲线坐标的连续函数。对于多边形曲线，双交替自联积分在几何概率方面获得了更简单的表达式。
In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.