Asymptotic laws for random knot diagrams
Asymptotic laws for random knot diagrams
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随机结图的渐近定律
DOI：
10.1088/1751-8121/aa6e45
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Harrison Chapman
中科院分区：
文献类型：
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作者：
Harrison Chapman
我们通过将结和链接图视为球体上装饰的（有根的）拓扑图来研究随机打结，并从给定数量的顶点 n 的集合中均匀地拉出它们，正如 Cantarella 和 Mastin 最近的工作中首次建立的那样。结图模型是一个令人兴奋的新模型，它捕获了打结空间曲线模型的随机几何形状以及从图中计算不变量的简便性。我们证明了结图是渐近指数罕见的，类似于萨姆纳和惠廷顿关于自回避多边形的里程碑式结果。我们的证明使用相同的关键思想：我们首先证明结图遵循模式定理，该定理描述了它们的分形结构。我们研究了这种行为在实践中发生的速度。因此，几乎所有图表都是不对称的，从而简化了该模型的采样。我们以该模型中打结的实验数据作为结论。这种随机打结模型与 Diao 等人和 Dunfield 等人研究的模型相似。
We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices n, as first established in recent work with Cantarella and Mastin. The knot diagram model is an exciting new model which captures both the random geometry of space curve models of knotting as well as the ease of computing invariants from diagrams. We prove that unknot diagrams are asymptotically exponentially rare, an analogue of Sumners and Whittington’s landmark result for self-avoiding polygons. Our proof uses the same key idea: we first show that knot diagrams obey a pattern theorem, which describes their fractal structure. We examine how quickly this behavior occurs in practice. As a consequence, almost all diagrams are asymmetric, simplifying sampling from this model. We conclude with experimental data on knotting in this model. This model of random knotting is similar to those studied by Diao et al, and Dunfield et al.