Average size of random polygons with fixed knot topology

2003/03/22 by Hiroshi Matsuda, Akihisa Yao, Hiroshi Tsukahara +3 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Algorithms and Data Compression #Combinatorics #Computational Geometry and Mesh Generation #Exponent #Geometric and Algebraic Topology #Geometry #Knot (papermaking) #Mathematics #Range (aeronautics) #Scaling #Topology (electrical circuits) #cond-mat.soft #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.68.011102

13 pages, 2 figures

arxiv created 2003/03/22 · openalex publication_date 2003/07/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We have evaluated by numerical simulation the average size R_K of random polygons of fixed knot topology K=\ensuremath∅,3₁,3₁\ensuremath\sharp4₁, and we have confirmed the scaling law R_K²\ensuremath∼N^2\ensuremathν_K for the number N of polygonal nodes in a wide range; N=100--2200. The best fit gives 2\ensuremathν_K\ensuremath≃1.11--1.16 with good fitting curves in the whole range of N. The estimate of 2\ensuremathν_K is consistent with the exponent of self-avoiding polygons. In a limited range of N (N\ensuremath\gtrsim600), however, we have another fit with 2\ensuremathν_K\ensuremath≃1.01--1.07, which is close to the exponent of random polygons.

Citations

• On the dominance of trivial knots among SAPs on a cubic lattice

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