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.

16 pages, v2, to appear in Algebraic and Geometric Topology. Major changes include removing Proposition 5.7 due to an error in the proof, generalizing the results to a larger class of actions on S^4, and the addition of a section discussing equivariant stabilization distance