All Prime Knots Through 10 Crossings Have Superbridge Index $\leq 5$

TL;DR

This paper establishes that all prime knots with up to 10 crossings have a superbridge index of at most 5 by providing new bounds on their stick numbers, completing the classification for these knots.

Contribution

It introduces new upper bounds on the stick numbers of specific 10-crossing knots, confirming their superbridge index does not exceed 5, thus completing the classification for prime knots up to 10 crossings.

Findings

All prime knots through 10 crossings have superbridge index ≤ 5

New bounds on stick numbers for several 10-crossing knots

Confirmed no prime knots through 10 crossings have superbridge index > 5

Abstract

This paper gives new upper bounds on the stick numbers of the knots $9_{18}$, $10_{18}$, $10_{58}$, $10_{66}$, $10_{68}$, $10_{80}$, $10_{82}$, $10_{84}$, $10_{93}$, $10_{100}$, and $10_{152}$, as well as on the equilateral stick number of $10_{79}$. These bounds imply that the knots $10_{58}$, $10_{66}$, and $10_{80}$ have superbridge index $\leq 5$, completing the project of showing that no prime knots through 10 crossings can have superbridge index larger than 5. The current best bounds on stick number and superbridge index for prime knots through 10 crossings are given in Appendix A.