On the genera of symmetric unions of knots

TL;DR

This paper establishes a relationship between the twisted Alexander polynomials of symmetric union knots and their partial knots, leading to genus inequalities and insights into their group epimorphisms, advancing understanding of ribbon knots.

Contribution

It introduces an identity linking twisted Alexander polynomials of symmetric unions and partial knots, and derives genus inequalities, addressing an open problem in knot theory.

Findings

Established an identity between twisted Alexander polynomials of symmetric unions and partial knots.

Derived genus inequalities for symmetric union knots.

Provided conditions to constrain symmetric union presentations of ribbon knots.

Abstract

In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.