Average size of random polygons with fixed knot topology

TL;DR

This study uses numerical simulations to analyze the average size of random polygons with fixed knot types, confirming a scaling law and comparing exponents with self-avoiding and random polygons.

Contribution

It provides the first detailed numerical analysis of how fixed knot topology influences the size scaling of random polygons.

Findings

Scaling law $R^2_K o N^{2 u_K}$ confirmed across a wide range of N

Exponent $2 u_K$ estimated between 1.11 and 1.16, aligning with self-avoiding polygons

In smaller N range, exponent close to 1, similar to random polygons

Abstract

We have evaluated by numerical simulation the average size $R_K$ of random polygons of fixed knot topology $K = \emptyset, 3_1, 3_1\sharp4_1$, and we have confirmed the scaling law $R^2_K \sim N^{2\nu_K}$ for the number $N$ of polygonal nodes in a wide range; $N = 100$ -- 2200. The best fit gives $2 \nu_K \simeq 1.11$ -- 1.16 with good fitting curves in the whole range of $N$. The estimate of $2 \nu_K$ is consistent with the exponent of self-avoiding polygons. In a limited range of $N$ ($N \gtrsim 600$), however, we have another fit with $2 \nu_K \simeq 1.01$ -- 1.07, which is close to the exponent of random polygons.