Symmetric ribbon disks

TL;DR

This paper investigates symmetric ribbon disks from symmetric union presentations of ribbon knots, introducing a new symmetric ribbon number, and demonstrates that the difference between this and the classical ribbon number can be arbitrarily large through specific knot constructions.

Contribution

It introduces the concept of symmetric ribbon number, compares it with the classical ribbon number, and constructs knots showing an unbounded gap between these measures.

Findings

The symmetric ribbon number can be arbitrarily larger than the classical ribbon number.

A simple description of symmetric unions in terms of band diagrams is provided.

An upper bound for the Heegaard genus of branched double covers of symmetric unions is established.

Abstract

We study the ribbon discs that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number is introduced and compared with the classical ribbon number. We show that the gap between these numbers can be arbitrarily large by constructing an infinite family of ribbon knots with ribbon number 2 and arbitrarily large symmetric ribbon number. 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.