A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting
Statistical Mechanics
2025-06-02 v3 Probability
Symplectic Geometry
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.
Keywords
Cite
@article{arxiv.2309.10163,
title = {A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting},
author = {Jason Cantarella and Henrik Schumacher and Clayton Shonkwiler},
journal= {arXiv preprint arXiv:2309.10163},
year = {2025}
}
Comments
12 pages, 4 figures. This version includes a table of estimated unknot probabilities for equilateral polygons from 16 to 3043 edges, and an additional data file "unknot-inverse-samples.tsv" giving the raw data from our experiment