Knot Probabilities in Random Diagrams

TL;DR

This paper calculates exact probabilities of different knot types in a random diagram model with up to 10 crossings, revealing that unknots dominate and follow a Zipf-like distribution.

Contribution

It provides the first comprehensive enumeration and probability analysis of knot types in a finite set of random diagrams, including the distribution pattern.

Findings

Most diagrams with 10 crossings are unknots (about 78%).

Tree-like diagrams are prevalent among unknots.

Knot probabilities follow a Zipf-like distribution.

Abstract

We consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to compute exact probabilities for knots in this model. As expected, most diagrams with 10 and fewer crossings are unknots (about 78% of the roughly 1.6 billion 10 crossing diagrams). For these crossing numbers, the unknot fraction is mostly explained by the prevalence of tree-like diagrams which are unknots for any assignment of over/under information at crossings. The data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type, analogous to Zipf's law for word frequency. All knot frequencies are available as ancillary data.