Symmetric ribbon disks

Abstract

We study the ribbon disks that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number rS(K) is introduced and compared with the classical ribbon number r(K). We show that the difference rS(K) - r(K) can be arbitrarily large by constructing an infinite family of ribbon knots Kn such that r(Kn) = 2 and rS(Kn) > n. The proof is based on a particularly simple description of symmetric unions in terms of certain band diagrams which leads to an upper bound for the Heegaard genus of their branched double covers.