New Upper Bounds for Stick Numbers

TL;DR

This paper employs simulated annealing with knot-preserving moves to find minimal stick number representations of knots, providing new bounds and exact values for many prime knots up to 13 crossings.

Contribution

It introduces a novel computational approach to determine upper bounds for knot stick numbers, achieving exact values for 19 previously unknown cases.

Findings

Established new upper bounds for prime knots up to 13 crossings.

Determined exact stick numbers for 19 knot types.

Provided a comprehensive table of bounds for all knots through 13 crossings.

Abstract

We use a version of simulated annealing with knot-type preserving moves to find polygonal representatives of various knot types with low stick number. These give better bounds on stick numbers of prime knots through 10 crossings, and for the first time give a comprehensive table of stick number bounds on all knots through 13 crossings. These are equal to existing lower bounds (and hence determine the stick number exactly) for 19 knot types whose exact stick number was not known previously.