标题： 随机绳结图的渐近定律 显示英文标题

标题： Asymptotic laws for random knot diagrams

摘要： 我们通过将纽结和链环图视为球面上的装饰（根）拓扑映射，研究随机纽结，并从给定顶点数的集合中均匀抽取$n$，如最近与Cantarella和Mastin的工作中首次确立的那样。 纽结图模型是一个令人兴奋的新模型，它既捕捉了空间曲线纽结模型的随机几何，又保留了从图中计算不变量的便利性。 我们证明了无纽结图在渐近意义上是指数级稀少的，这是Sumners和Whittington对于非交叉行走的开创性结果的类比。 我们的证明使用了相同的关键思想：我们首先证明纽结图遵守一个模式定理，该定理描述了它们的分形结构。 我们研究这种行为在实际中发生得多快。 作为结果，几乎所有的图都是不对称的，这简化了从该模型中进行采样。 我们最后给出了该模型中纽结的实验数据。 这种随机纽结模型与Diao等人和Dunfield等人研究的模型类似。 显示英文摘要

摘要： 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 walks. 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.