Invariants of Random Knots and Links

TL;DR

This paper analyzes the statistical properties of random knots and links in three-dimensional space using the Petaluma model, providing explicit formulas for invariant distributions and comparing them with other models.

Contribution

It introduces the first precise formulas for the distributions of invariants in any random knot or link model, specifically for linking number, Casson invariant, and v3.

Findings

Derived formulas for the distribution of linking number in the Petaluma model.

Calculated expectations and higher moments of Casson invariant and v3.

Numerical comparisons with other random knot and link models.

Abstract

We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3. These are the first precise formulas given for the distributions of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams.