Equivariant Double-Slice Genus, Stabilization, and Equivariant Stabilization

TL;DR

This paper introduces the concepts of equivariant double-slice and super-slice genus for strongly invertible knots, providing lower bounds and applications to symmetric surfaces and stabilization distances.

Contribution

It defines new equivariant genera for knots, establishes lower bounds, and applies these to problems in symmetric surface topology and stabilization distances.

Findings

Found a family of knots with high equivariant double-slice genus

Constructed unknotted symmetric 2-spheres not bounding symmetric 3-balls

Provided bounds for 1-handle stabilization distance

Abstract

In this paper we define the equivariant double-slice genus and equivariant super-slice genus of a strongly invertible knot. We prove lower bounds for both the equivariant double-slice genus and the equivariant super-slice genus. Using these bounds we find a family of knots which are double-slice and equivariantly slice, but have equivariant double-slice genus at least $n$. Using this result, we construct unknotted symmetric 2-spheres which do not bound symmetric 3-balls. Additionally, using double-slice and super-slice genera we find effective lower bounds for 1-handle stabilization distance and identify a possible method for using equivariant double-slice and super-slice genera to bound symmetric 1-handle stabilization distance for symmetric surfaces.