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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >Sampling large random knots in a confined space
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Sampling large random knots in a confined space

机译：在有限的空间中采样大的随机结

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摘要

DNA knots formed under extreme conditions of condensation, as in bacteriophage P4, are difficult to analyze experimentally and theoretically. In this paper, we propose to use the uniform random polygon model as a supplementary method to the existing methods for generating random knots in confinement. The uniform random polygon model allows us to sample knots with large crossing numbers and also to generate large diagrammatically prime knot diagrams. We show numerically that uniform random polygons sample knots with large minimum crossing numbers and certain complicated knot invariants (as those observed experimentally). We do this in terms of the knot determinants or colorings. Our numerical results suggest that the average determinant of a uniform random polygon of n vertices grows faster than O(e(n2)). We also investigate the complexity of prime knot diagrams. We show rigorously that the probability that a randomly selected 2D uniform random polygon of n vertices is almost diagrammatically prime goes to 1 as n goes to infinity. Furthermore, the average number of crossings in such a diagram is at the order of O(n(2)). Therefore, the two-dimensional uniform random polygons offer an effective way in sampling large (prime) knots, which can be useful in various applications.
机译：如在噬菌体P4中一样，在极端冷凝条件下形成的DNA结很难通过实验和理论进行分析。在本文中，我们建议使用统一的随机多边形模型作为现有方法的补充方法，以限制现有的随机结。统一的随机多边形模型使我们可以对具有较大交叉数的结进行采样，并且还可以生成大型图解质结图。我们从数值上显示均匀的随机多边形对具有大的最小交叉数和某些复杂的结不变性（如实验观察到的）的结进行采样。我们根据结点决定因素或颜色进行此操作。我们的数值结果表明，n个顶点的均匀随机多边形的平均行列式比O（e（n2））增长更快。我们还研究了素结图的复杂性。我们严格地表明，随着n趋于无穷大，随机选择的n个顶点的2D均匀随机多边形几乎以图形方式表示的质数的概率变为1。此外，这种图中的平均交叉次数为O（n（2））。因此，二维均匀随机多边形为采样大的（原始）结提供了一种有效的方法，这在各种应用中都非常有用。

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