Ribbon numbers of 12-crossing knots

TL;DR

This paper investigates the ribbon number of 12-crossing knots, analyzing Alexander polynomials to compute ribbon numbers and exploring the limitations of polynomial invariants in higher-genus cases.

Contribution

It extends the understanding of ribbon numbers by systematically analyzing Alexander polynomials and their relation to ribbon genus beyond genus zero.

Findings

Alexander polynomials of knots with ribbon number ≤4 include 56 polynomials

Computed ribbon numbers for many 12-crossing knots

Higher-genus ribbon numbers show complex behavior, not fully controlled by Alexander polynomial

Abstract

The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and Zupan. We show that the set of Alexander polynomials of knots with ribbon number at most four contains 56 polynomials, and we use this set to compute the ribbon numbers for many 12-crossing knots. We also study higher-genus ribbon numbers of knots, presenting some examples that exhibit interesting behavior and establishing that the success of the Alexander polynomial at controlling genus-0 ribbon numbers does not extend to higher genera.