Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either “local” or “non-local”, based on a standard approach for measuring the “size” of a knot within a knotted polymer chain. The method involves associating knot-types to subarcs of the chain, and then identifying a knotted subarc with minimal arclength; this arclength is then the knot-size. If the resulting knot-size is small relative to the whole length of the chain, then the knot is considered to be localised or “local”; otherwise, it is “non-local”. Using this definition, we establish that all but exponentially few sufficiently long self-avoiding polygons (closed chains) in a tubular sublattice of the simple cubic lattice are “non-locally” knotted. This is shown to also hold for the case when the same polygons are subject to an external tensile force, as well as in the extreme case when they are as compact as possible (no empty lattice sites). We also provide numerical evidence for small tube sizes that at equilibrium non-local knotting is more likely than local knotting, regardless of the strength of the stretching or compressing force. The relevance of these results to other models and recent experiments involving DNA knots is also discussed.

中文翻译：

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表征纳米通道中聚合物的打结特性†

使用测量管中聚合物的晶格模型，我们基于一种标准方法来测量给定结的不同配置为“局部”或“非局部”的一种方法，该方法用于测量结中“结”的大小。打结的聚合物链。该方法包括将打结类型与链的子弧相关联，然后识别具有最小弧长的打结的子弧。然后，此弧长为结大小。如果结的大小相对于链条的整个长度较小，则该结被认为是局部的或“局部的”。否则，它是“非本地”的。使用此定义，我们确定在简单立方晶格的管状子晶格中，除指数增长外几乎所有的足够长的自避免多边形（闭合链）都被“非局部”打结了。对于相同的多边形在受到外部拉力的情况下，以及在它们尽可能紧凑（无空晶格位置）的极端情况下，这也显示成立。我们还为小尺寸的管提供了数值证据，表明在平衡状态下，非局部打结比局部打结的可能性更大，而与拉伸或压缩力的强度无关。还讨论了这些结果与其他模型以及涉及DNA结的最新实验的相关性。