Abstract

A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this article is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sampling, however that method is inefficient at even modest sizes. We introduce Markov chains that sample from the space of plane curves using local moves based on Reidemeister moves. By then mapping vertices on those curves to crossings we produce random knot diagrams. Combining this chain with flat histogram methods we achieve an efficient sampler of plane curves and knot diagrams. By analyzing data from this chain we are able to estimate the number of knot diagrams of a given size and also compute knotting probabilities and so investigate their asymptotic behavior.

摘要

平面曲线是一个节点图，其中每个交叉点都被一个 4 价顶点替换，因此与平面四边形的子集是对偶的。本文的目的是介绍一种通过平面曲线采样来采样图表的新工具。目前，最有效的图表抽样方法是拒绝抽样，但是该方法即使在适度的规模下也效率低下。我们介绍了使用基于 Reidemeister 移动的局部移动从平面曲线空间中采样的马尔可夫链。然后将这些曲线上的顶点映射到交叉点，我们生成随机节点图。将此链与平面直方图方法相结合，我们实现了平面曲线和节点图的有效采样器。