Equivariantly Slice Knots in Symmetric 4-Manifolds

TL;DR

This paper investigates the equivariant 4-genus of strongly invertible knots in symmetric 4-manifolds, introducing new techniques for constructing slice disks and demonstrating differences from standard 4-genus.

Contribution

It develops an equivariant tubing construction method and shows that the equivariant 4-genus can differ from the standard 4-genus and other equivariant versions.

Findings

The equivariant 4-genus can differ from the standard 4-genus.

Constructed slice disks for knots in symmetric 4-manifolds.

Example showing the figure 8 knot is equivariantly slice in S^2×S^2 with one involution but not the other.

Abstract

We study the equivariant 4-genus of strongly invertible knots in the $S^3$ boundary of 4-manifolds with involution. We provide techniques for constructing slice disks for knots in various symmetric 4-manifolds via an equivariant version of Marengon and Mihajlovi\`c's tubing construction. Using these techniques, we show that this equivariant 4-genus can differ from the standard 4-genus function of the 4-manifold as well as the equivariant 4-genus of $S^4$. As an example, we show that $S^2\times S^2$ admits an involution such that the figure $8$ knot is equivariantly slice with respect to one of its two strong inversions but not the other.