Bounding the ribbon numbers of knots and links

TL;DR

This paper establishes new lower bounds for the ribbon number of knots and links using polynomial invariants, proves finiteness of certain sets of Alexander polynomials, and computes ribbon numbers for knots with up to 11 crossings.

Contribution

It introduces novel bounds for ribbon numbers based on Alexander polynomials and determines the sets of polynomials for small ribbon numbers, advancing understanding of knot complexity.

Findings

Derived new lower bounds for ribbon numbers using et(K) and elta_K(t).

Proved the finiteness and computability of the set _r of Alexander polynomials for knots with ribbon number  r.

Computed ribbon numbers for all ribbon knots with 11 or fewer crossings, with three exceptions.

Abstract

The ribbon number $r(K)$ of a ribbon knot $K \subset S^3$ is the minimal number of ribbon intersections contained in any ribbon disk bounded by $K$. We find new lower bounds for $r(K)$ using $\det(K)$ and $\Delta_K(t)$, and we prove that the set $\mathfrak{R}_r~=~\{\Delta_K(t)~:~r(K)~\leq~r\}$ is finite and computable. We determine $\mathfrak{R}_2$ and $\mathfrak{R}_3$, applying our results to compute the ribbon numbers for all ribbon knots with 11 or fewer crossings, with three exceptions. Finally, we find lower bounds for ribbon numbers of links derived from their Jones polynomials.