Equivariant Q-sliceness of strongly invertible knots

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.