New Computations of the Superbridge Index

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.