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Properties of ideal composite knots

机译：理想的复合结的特性

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The shortest tube of constant diameter that can form a given knot represents the 'ideal' form of the knot. Ideal knots provide an irreducible representation of the knot, and they have some intriguing mathematical and physical features, including a direct correspondence with the time-averaged shapes of knotted DNA molecules in solution. Here we describe the properties of ideal forms of composite knots-knots obtained by the sequential tying of two or more independent knots (called factor knots) on the same string. We find that the writhe (related to the handedness of crossing points) of composite knots is the sum of that of the ideal forms of the factor knots. By comparing ideal composite knots with simulated configurations of knotted, thermally fluctuating DNA, we conclude that the additivity of writhe applies also to randomly distorted configurations of composite knots and their corresponding factor knots. We show that composite knots with several factor knots may possess distinct structural isomers that can be interconverted only by loosening the knot.
机译：可以形成给定结的直径最小的恒定管代表结的“理想”形式。理想的结提供了无法还原的结表示，并且它们具有一些有趣的数学和物理特征，包括与溶液中打结的DNA分子的时间平均形状的直接对应。在这里，我们描述了通过在同一根弦上顺序绑扎两个或多个独立结（称为因子结）而获得的复合结的理想形式的特性。我们发现复合结的扭结（与交叉点的手性有关）是因子结的理想形式的总和。通过比较理想的复合材料打结和模拟的打结，热波动的DNA构型，我们得出结论，线粒的可加性也适用于复合材料打结及其相应因子打结的随机扭曲构型。我们显示具有多个因子结的复合结可能具有独特的结构异构体，这些结构异构体仅通过松开该结即可相互转换。

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