A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting

TL;DR

This paper introduces a faster algorithm for sampling random equilateral closed polygons in 3D and uses it to study the probability of knotting, specifically unknots, in such polygons.

Contribution

A novel, more efficient sampling algorithm for equilateral closed polygons and a new computational approach for knot invariant analysis.

Findings

Improved sampling speed with quadratic time complexity.

New insights into the probability of unknots in equilateral polygons.

Enhanced computational tools for knot detection.

Abstract

We present a faster direct sampling algorithm for random equilateral closed polygons in three-dimensional space. This method improves on the moment polytope sampling algorithm of Cantarella, Duplantier, Shonkwiler, and Uehara (2016) and has (expected) time per sample quadratic in the number of edges in the polygon. We use our new sampling method and a new code for computing invariants based on the Alexander polynomial to investigate the probability of finding unknots among equilateral closed polygons.