论文标题
典型的结：尺寸，链接组件计数和writhe
Typical knots: size, link component count, and writhe
论文作者
论文摘要
我们使用网格图对结和链接的典型行为进行建模。在科学中，联系无处不在，它们的“正常”或“典型”行为对于理解情况（例如DNA的拓扑状态或环形聚合物的统计力学）至关重要。我们检查了三个不变的：随机结的预期大小；随机链接的预期组件数量；以及一个随机结的预期扭曲。我们在数值上研究了前两个，并产生生成的函数，这些函数将观察到的模式编纂：结尺寸均匀分布并线性依赖于网格大小，并且组件数量遵循一个分布的分布，其平均值和方差随网格大小的log_2增长；特别是，对于任何固定的K，随着网格大小的增加，K分量链接变得消失了。最后，我们观察到Writhe的奇数矩消失了，我们进行了探索性数据分析，发现方差随着网格大小和峰度的正方形而增长，恒定约为3.5。我们将在未来的工作中继续进行该项目，我们研究属及其对跨的影响。
We model the typical behavior of knots and links using grid diagrams. Links are ubiquitous in the sciences, and their "normal" or "typical" behavior is of significant importance in understanding situations such as the topological state of DNA or the statistical mechanics of ring polymers. We examine three invariants: the expected size of a random knot; the expected number of components of a random link; and the expected writhe of a random knot. We investigate the first two numerically and produce generating functions which codify the observed patterns: knot size is uniformly distributed and linearly dependent upon grid size, and the number of components follows a distribution whose mean and variance grow with log_2 of grid size; in particular, for any fixed k, the k-component links grow vanishingly rare as grid size increases. Finally, we observe that the odd moments of writhe vanish, and we perform an exploratory data analysis to discover that variance grows with the square of grid size and kurtosis is constant at approximately 3.5. We continue this project in a future work, where we investigate genus and the effects of crossing change on it.
