Knot complexity and the probability of random knotting

TL;DR

This paper investigates how the likelihood of a random polygon forming a specific knot type decreases exponentially with the knot's complexity, using computer simulations and knot invariants.

Contribution

It introduces a model linking knot complexity measures, like minimal crossing number and aspect ratio, to the probability of knot formation in random polygons.

Findings

Knotting probability decreases exponentially with knot complexity.

Knot complexity can be quantified by minimal crossing number and aspect ratio.

Simulation results support the exponential decay relationship.

Abstract

The probability of a random polygon (or a ring polymer) having a knot type $K$ should depend on the complexity of the knot $K$. Through computer simulation using knot invariants, we show that the knotting probability decreases exponentially with respect to knot complexity. Here we assume that some aspects of knot complexity are expressed by the minimal crossing number $C$ and the aspect ratio $p$ of the tube length to the diameter of the {\it ideal knot} of $K$, which is a tubular representation of $K$ in its maximally inflated state.