Equivariant double-slice genus, stabilization, and equivariant stabilization

Gabbard, Malcolm

doi:10.2140/agt.2025.25.5137

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Equivariant double-slice genus, stabilization, and equivariant stabilization

Malcolm Gabbard

Algebraic & Geometric Topology 25 (2025) 5137–5152

DOI: 10.2140/agt.2025.25.5137

Abstract

We define the equivariant double-slice genus and equivariant superslice genus of a strongly invertible knot. We prove lower bounds for both the equivariant double-slice genus and the equivariant superslice 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 superslice genera we find effective lower bounds for 1-handle stabilization distance and identify a possible method for using equivariant double-slice and superslice genera to bound the symmetric 1-handle stabilization distance for symmetric surfaces.

Keywords

slice genus, double-slice genus, symmetric knots, stabilization, knotted surfaces

Mathematical Subject Classification

Primary: 57K10, 57K45

Secondary: 57K40

References

Publication

Received: 27 May 2024

Revised: 10 October 2024

Accepted: 2 November 2024

Published: 20 November 2025

Authors

Malcolm Gabbard
Kansas State University Manhattan, KS United States

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Volume 25, issue 8 (2025)