Equivariant Q-sliceness of strongly invertible knots

TL;DR

This paper introduces the concept of equivariant $Q$-sliceness for strongly invertible knots, providing constructive examples, obstructions, and exploring the related equivariant $Q$-concordance group.

Contribution

It defines equivariant $Q$-sliceness, proves that Klein amphichiral knots are equivariant $Q$-slice, and extends classical obstructions to this setting.

Findings

Klein amphichiral knots are equivariant $Q$-slice in a $Q$-homology 4-ball.

The equivariant Fox-Milnor condition obstructs equivariant $Q$-sliceness.

Introduction of the equivariant $Q$-concordance group and analysis of natural maps.

Abstract

We introduce and study the notion of equivariant $\mathbb{Q}$-sliceness for strongly invertible knots. On the constructive side, we prove that every Klein amphichiral knot, which is a strongly invertible knot admitting a compatible negative amphichiral involution, is equivariant $\mathbb{Q}$-slice in a single $\mathbb{Q}$-homology $4$-ball, by refining Kawauchi's construction and generalizing Levine's uniqueness result. On the obstructive side, we show that the equivariant version of the classical Fox-Milnor condition, proved recently by the first author, also obstructs equivariant $\mathbb{Q}$-sliceness. We then introduce the equivariant $\mathbb{Q}$-concordance group and study the natural maps between concordance groups as an application. We also list some open problems for future study.