New Stick Number Bounds from Random Sampling of Confined Polygons

TL;DR

This paper uses large-scale Monte Carlo simulations of confined polygons to improve bounds on the stick number of knots, providing new data for knots with up to 10 crossings.

Contribution

It introduces a Monte Carlo method to generate large ensembles of random polygons, yielding new bounds on the stick number for many knots previously lacking precise data.

Findings

Generated 220 billion random polygons to estimate or bound stick numbers.

Improved bounds for over 40% of knots with 10 or fewer crossings.

Provided the most comprehensive bounds to date for these knots.

Abstract

The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds on the stick number. We adopt a Monte Carlo approach to finding better bounds, producing very large ensembles of random polygons in tight confinement to look for new examples of knots constructed from few segments. We generated a total of 220 billion random polygons, yielding either the exact stick number or an improved upper bound for more than 40% of the knots with 10 or fewer crossings for which the stick number was not previously known. We summarize the current state of the art in Appendix A, which gives the best known bounds on stick number for all knots up to 10 crossings.