Models of Random Knots

TL;DR

This paper reviews probabilistic models of random knots, their properties, and recent results, highlighting applications in natural sciences and exploring invariants and universality in knot distributions.

Contribution

It introduces and analyzes several known and new randomized models of knots, including the Petaluma model, with a focus on their properties and invariants.

Findings

Distribution of knot invariants in different models

Asymptotic behavior of invariants in the Petaluma model

Numerical and rigorous results on knot properties

Abstract

The study of knots and links from a probabilistic viewpoint provides insight into the behavior of "typical" knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.