Asymptotic laws for random knot diagrams

TL;DR

This paper introduces a new model for random knot diagrams, proving that unknots are exponentially rare and demonstrating their fractal structure, with implications for sampling and understanding knot complexity.

Contribution

It establishes the asymptotic rarity of unknots in the diagram model and proves a pattern theorem showing fractal structure in random knot diagrams.

Findings

Unknots are exponentially rare among large diagrams.

Most diagrams are asymmetric, aiding sampling.

Knot diagrams exhibit fractal patterns described by the pattern theorem.

Abstract

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.