Random knots using Chebyshev billiard table diagrams

TL;DR

This paper introduces a probabilistic model for generating random knots using Chebyshev billiard table diagrams, analyzes the likelihood of knot types, and conjectures that complex knots become rare as crossing numbers increase.

Contribution

It provides a new random knot model based on Chebyshev diagrams and derives formulas for knot probabilities within this framework.

Findings

Probability of selecting a specific knot decreases as crossings increase

Formulas for knot probabilities in the bridge index ≤ 2 class

Numerical evidence supports the decay of knot occurrence with more crossings

Abstract

We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot appearing in this model decays to zero as the number of crossings goes to infinity.