A faster direct sampling algorithm for equilateral closed polygons and the probability of knotting (Q6572806)

The authors present a faster algorithm for direct sampling of random equilateral closed polygons in three-dimensional space. This method improves on the moment polytope sampling algorithm of \textit{J. Cantarella} et al. [J. Phys. A, Math. Theor. 49, No. 27, Article ID 275202, 9 p. (2016; Zbl 1342.82063)] which achieved a runtime of \(\Theta (n^{5/2})\) per sample, where \(n\) is the number of edges in the random equilateral closed polygon. The improved algorithm presented here improves the runtime to \(\Theta (n^{2})\) per sample, The authors use the new sampling method to investigate the probability of finding unknots among equilateral closed polygons covering a polygonal length from \(n = 16\) to \(n = 3043\). The authors detect the unknot by computing invariants based on the Alexander polynomial.