Equivalence of symmetric union diagrams

TL;DR

This paper investigates symmetric union diagrams, a construction related to ribbon knots, by extending Reidemeister moves to symmetric cases and exploring the uniqueness and diversity of symmetric union representations.

Contribution

It introduces a framework of symmetric Reidemeister moves to analyze symmetric union diagrams and demonstrates the existence of multiple symmetric union representations for certain ribbon knots.

Findings

Every symmetric union diagram represents a ribbon knot.

An infinite family of ribbon two-bridge knots admits multiple symmetric union representations.

Abstract

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of uniqueness. In order to attack this question we extend the usual Reidemeister moves to a family of moves respecting the symmetry, and consider the symmetric equivalence thus generated. This notion being in place, we discuss several situations in which a knot can have essentially distinct symmetric union representations. We exhibit an infinite family of ribbon two-bridge knots each of which allows two different symmetric union representations.