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

Clayton Shonkwiler; Henrik Schumacher; Jason Cantarella

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

Jason Cantarella, Henrik Schumacher, Clayton Shonkwiler

Statistical Mechanics · Submitted 18 Sep 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We present a faster direct sampling algorithm for random equilateral closed polygons in three-dimensional space.

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arXiv ID2309.10163 (version 3)

Pith ID7H7YFTVN, Pith's permanent record of this paper and its review

CitationClayton Shonkwiler, Henrik Schumacher, Jason Cantarella. "A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting." arXiv:2309.10163 (2023). https://pith.science/paper/7H7YFTVN

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,985 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.10 · human participants 0.02 · controlled comparison 0.20 · randomized assignment 0.02 · states quantitative result 0.32 · compares with prior results 0.95 · mentions code or data availability 0.06.

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a faster direct sampling algorithm for random equilateral closed polygons in three-dimensional space.	contribution 0.99	0.96	0.25	asserted 1.00
s1	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.	contribution 0.62	0.77	0.17	asserted 1.00
s2	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.	contribution 0.72	0.94	0.19	asserted 0.89

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.99.

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arXiv listing abstract v3

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Original 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.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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The resulting method is faster and more general than the Action-Angle method described in [ref] .

CoBarS: Fast reweighted sampling for polygon spaces in any dimension · Introduction · 1.00

Cites it as background 1 paper

This is the point of view taken by mathematicians who study polygons in $^3$ via their symplectic structure [ref] , and by the present authors in our previous papers [ref] .

CoBarS: Fast reweighted sampling for polygon spaces in any dimension · The quotient by $(d)$ · 0.80

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