Invariant Seifert surfaces for strongly invertible knots

TL;DR

This paper investigates invariant Seifert surfaces for strongly invertible knots, showing that the difference between equivariant and usual genus can be arbitrarily large, contrasting with Edmonds' theorem for periodic knots.

Contribution

It proves that the gap between equivariant and usual genus can be arbitrarily large for strongly invertible knots and provides variants of Edmonds' theorem relevant to invariant Seifert surfaces.

Findings

The gap between equivariant and usual genus can be arbitrarily large.

Variants of Edmonds' theorem are established for strongly invertible knots.

Invariant Seifert surfaces exhibit significant genus differences in this context.

Abstract

We study invariant Seifert surfaces for strongly invertible knots, and prove that the gap between the equivariant genus (the minimum of the genera of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genus of the underlying knot can be arbitrary large. This forms a sharp contrast with Edmonds' theorem that every periodic knot admits an invariant minimal genus Seifert surface. We also prove variants of Edmonds' theorem, which are useful in studying invariant Seifert surfaces for strongly invertible knots.