Equivariant Double-Slice Genus, Stabilization, and Equivariant Stabilization

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.