An ergodic algorithm for generating knots with a prescribed injectivity radius

TL;DR

This paper introduces an ergodic algorithm for generating thick equilateral knots with a prescribed injectivity radius, enabling analysis of how thickness influences geometric properties like radius of gyration.

Contribution

It presents the first ergodic sampling algorithm for thick knots, ensuring connectivity between any two knots under thickness constraints and improving generation speed.

Findings

Radius of gyration increases with thickness

Growth exponent for radius of gyration increases with thickness

Algorithm outperforms previous methods in speed

Abstract

The first algorithm for sampling the space of thick equilateral knots, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. To prove the existence of the algorithm, we describe a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This approach ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint and so is ergodic. This ergodic sampling unlocks the ability to analyze the effects of thickness on properties of the geometric knot such as radius of gyration. This algorithm will be shown to be faster than previous methods for generating thick knots, and the data from this algorithm shows that the radius of gyration increases strongly with thickness and that the growth exponent for radius of gyration increases with thickness.