Geometry and physics of knots
Geometry and physics of knots
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DOI：
10.1038/384142a0
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发表时间：
1996-11-14
期刊：
影响因子：
64.8
通讯作者：
Stasiak, A
中科院分区：
文献类型：
--
作者：
Katritch, V;Bednar, J;Stasiak, A
结通常根据结的空间配置变化下不变的拓扑特性进行分类（1-4）。在这里，我们从一个不同的角度来处理结识别，通过考虑我们定义为“理想”的特定几何形式的属性。对于具有给定拓扑结构并由均匀直径的管组装的结，理想形式是具有最高体积与表面积比的几何构型。实际上，这相当于确定可以闭合以形成结的最短管段。由于理想形式的概念与绝对空间尺度无关，因此提供理想表示的管的长径比是恒定的，而与管的实际尺寸无关。我们报告了计算机模拟的结果，表明这些结的理想表示具有令人惊讶的简单几何特性。特别是，有一个简单的线性关系之间的长度与直径比和交叉数的数量在二维投影的结平均在所有方向上的交叉。我们还发现，在热平衡时，打结的高分子链的平均形状与相应的结型的理想表示密切相关。我们的观察提供了理想几何物体和看似无序系统的行为之间的联系，并允许预测打结聚合物的性质，如它们的电泳迁移率（5）。
KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration(1-4). Here we approach knot identification from a different angle, by considering the properties of particular geometrical forms which we define as 'ideal'. For a knot with a given topology and assembled from a tube of uniform diameter, the ideal form is the geometrical configuration having the highest ratio of volume to surface area. Practically, this is equivalent to determining the shortest piece of tube that can be closed to form the knot. Because the notion of an ideal form is independent of absolute spatial scale, the length-to-diameter ratio of a tube providing an ideal representation is constant, irrespective of the tube's actual dimensions. We report the results of computer simulations which show that these ideal representations of knots have surprisingly simple geometrical properties. In particular, there is a simple linear relationship between the length-to-diameter ratio and the crossing number-the number of intersections in a two-dimensional projection of the knot averaged over all directions. We have also found that the average shape of knotted polymeric chains in thermal equilibrium is closely related to the ideal representation of the corresponding knot type. Our observations provide a link between ideal geometrical objects and the behaviour of seemingly disordered systems, and allow the prediction of properties of knotted polymers such as their electrophoretic mobility(5).