New Stick Number Bounds from Random Sampling of Confined Polygons

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.

Keywords:

• Knot theory
• Monte Carlo methods
• Random polygons
• Stick number

Acknowledgments

We are grateful to Colin Adams, Ryan Blair, Jason Cantarella, Harrison Chapman, Tetsuo Deguchi, Kate Hake, Gyo Taek Jin, Ken Millett, Eric Rawdon, Erica Uehara, and the anonymous referees for helpful comments, suggestions, and conversations about random polygons and stick knots.

Declaration of interest

No potential conflict of interest was reported by the author(s).

Notes

1 For knots up to 10 crossings we use Alexander–Briggs notation consistent with the (Perko-corrected) Rolfsen table [53]; for knots with 11 or more crossings, we use the Dowker–Thistlethwaite naming convention.

2 At least in high dimensions, a single hit-and-run step is unlikely to move very far, so it is preferable to do several steps at once, though not too many since hit-and-run steps are much more expensive than sampling from the torus. Our experience is that 10 steps provides a good balance.

3 The best theoretical result along these lines is due to Hake [26], who proved that the probability that a random equilateral hexagon is knotted is bounded above by $\frac{14-3\pi }{192}<\frac{1}{42}$; empirically, the fraction of nontrivial knots among random hexagons is close to $\frac{1}{10,000}$.

4 As described above, we use $\beta =\frac{1}{2}$, so that hit-and-run steps and torus samples are equally likely, and γ = 10, so that we iterate hit-and-run 10 times whenever we choose to do a hit-and-run step.

5 “Unfortunately, the diagrams of ${10}_{83}$ and ${10}_{86}$ given by Rolfsen do not correspond to their Conway notation or Alexander polynomial—they are interchanged” [27]. This was only corrected in the 2003 edition of Rolfsen’s book—after Rawdon and Scharein’s article—so it is not surprising that our pipeline identified Rawdon and Scharein’s ${10}_{83}$ as a ${10}_{86}$ and identified their ${10}_{86}$ as a ${10}_{83}$. This is an issue that researchers in the broader computational knot theory community should be aware of: for example, the invariants computed by SnapPy for ${10}_{83}$ match the invariants in the Knot Atlas [38], pyknotid, and KnotInfo [18] for ${10}_{86}$, and conversely the SnapPy invariants for ${10}_{86}$ agree with the Knot Atlas, pyknotid, and KnotInfo invariants for ${10}_{83}$, so whether you identify a given knot as ${10}_{83}$ or ${10}_{86}$ may depend on which software you use. We have used the Knot Atlas naming convention in this article.

Additional information

Funding

This work was partially supported by a grant from the Simons Foundation (#354225, CS).