A Markov Chain Sampler for Plane Curves

TL;DR

This paper introduces an efficient Markov chain-based method for sampling plane curves and knot diagrams, enabling better estimation of their quantities and properties compared to traditional rejection sampling.

Contribution

It presents a novel Markov chain approach using Reidemeister moves for sampling plane curves and knot diagrams, improving efficiency and enabling statistical analysis.

Findings

Efficient sampling of plane curves and knot diagrams achieved.

Estimated number of knot diagrams of given size provided.

Analyzed knotting probabilities and asymptotic behavior.

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 paper 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 analysing 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 behaviour.