New Computations of the Superbridge Index

TL;DR

This paper presents new upper bounds for the superbridge index of various knots, significantly expanding the known cases from 29 to 49, and provides updated knowledge on prime knots up to 10 crossings.

Contribution

It introduces novel upper bounds on the superbridge index that are not derived from stick number, increasing the known cases from 29 to 49.

Findings

Superbridge index of several knots is equal to 4.

Number of knots with known superbridge index increased from 29 to 49.

Updated overview of superbridge index for prime knots up to 10 crossings.

Abstract

The knots $8_1$, $8_2$, $8_3$, $8_5$, $8_6$, $8_7$, $8_8$, $8_{10}$, $8_{11}$, $8_{12}$, $8_{13}$, $8_{14}$, $8_{15}$, $9_7$, $9_{16}$, $9_{20}$, $9_{26}$, $9_{28}$, $9_{32}$, and $9_{33}$ all have superbridge index equal to 4. This follows from new upper bounds on superbridge index not coming from the stick number and increases the number of knots from the Rolfsen table for which superbridge index is known from 29 to 49. Appendix A gives the current state of knowledge of superbridge index for prime knots through 10 crossings.