Slice knots which bound punctured Klein bottles

TL;DR

This paper studies knots in S^3 bounding Klein bottles with zero framing, establishing conditions for sliceness in Z[1/2]-homology 4-balls and exploring cable knot concordance related to slice properties.

Contribution

It proves that a knot bounding a Klein bottle with zero framing is slice in a Z[1/2]-homology 4-ball if and only if its core J is, and relates cable knot concordance to sliceness in this context.

Findings

Knot J has zero self-linking if K bounds a Klein bottle with zero framing.

Knot K is slice in a Z[1/2]-homology 4-ball iff J is.

(2, p) cables of K and J are Z[1/2]-concordant iff K and J are.

Abstract

We investigate the properties of knots in S^3 which bound Klein bottles, such that a pushoff of the knot has zero linking number with the knot, i.e. has zero framing. This is motivated by the many results in the literature regarding slice knots of genus one, for example, the existence of homologically essential zero self-linking simple closed curves on genus one Seifert surfaces for algebraically slice knots. Given a knot K bounding a Klein bottle F with zero framing, we show that J, the core of the orientation-preserving band in any disk-band form of F, has zero self-linking. We prove that such a K is slice in a Z[1/2]-homology 4-ball if and only if J is as well, a stronger result than what is currently known for genus one slice knots. As an application, we prove that given knots K and J and any odd integer p, the (2, p) cables of K and J are Z[1/2]-concordant if and only if K and J are Z[1/2]-concordant. In particular, if the (2,1)-cable of a knot K is slice, K is slice in a Z[1/2]-homology ball.