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.

中文翻译：

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开曲线和闭曲线的结多项式

在本手稿中，我们介绍了一种测量 3 空间中曲线纠缠的方法，该方法将节点和链接多项式的概念扩展到开曲线。我们在 3 空间中定义曲线的括号多项式，并表明它具有实系数并且是曲线坐标的连续函数。这用于定义琼斯多项式，使其适用于 3 空间中的开曲线和闭曲线。对于开式曲线，琼斯多项式具有实系数，并且是曲线坐标的连续函数，并且由于曲线的端点趋于重合，因此开式曲线的琼斯多项式趋向于结果节点的琼斯多项式。对于闭合曲线，它是拓扑不变量，就像经典的琼斯多项式一样。