New Superbridge Index Calculations from Non-Minimal Realizations

TL;DR

This paper extends polygonal realization techniques to compute the superbridge index of knots with odd edges, determining exact indices for many new knots and revealing instances where minimal superbridge realizations do not minimize stick number.

Contribution

It introduces an extension of superbridge index calculation methods to odd-edged polygonal realizations, identifying new exact indices for numerous knots and providing comprehensive knot data.

Findings

Exact superbridge indices for many 9-crossing knots

First determination of superbridge indices for several 12-crossing knots

Discovery of superbridge-minimizing realizations that do not minimize stick number

Abstract

Previous work used polygonal realizations of knots to reduce the problem of computing the superbridge number of a realization to a linear programming problem, leading to new sharp upper bounds on the superbridge index of a number of knots. The present work extends this technique to polygonal realizations with an odd number of edges and determines the exact superbridge index of many new knots, including the majority of the 9-crossing knots for which it was previously unknown and, for the first time, several 12-crossing knots. Interestingly, at least half of these superbridge-minimizing polygonal realizations do not minimize the stick number of the knot; these seem to be the first such examples. Appendix A gives a complete summary of what is currently known about superbridge indices of prime knots through 10 crossings and Appendix B gives all knots through 16 crossings for which the superbridge index is known.