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 . 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.
Keywords
Cite
@article{arxiv.2404.17062,
title = {Equivariant Double-Slice Genus, Stabilization, and Equivariant Stabilization},
author = {Malcolm Gabbard},
journal= {arXiv preprint arXiv:2404.17062},
year = {2025}
}
Comments
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