Bounding the ribbon numbers of knots and links

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 ${\mathrm{\Delta }}_K(t)$, and we prove that the set ${\mathfrak{R}}_r=\{{\mathrm{\Delta }}_K(t):r(K)\le r\}$ 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.