Slice knots which bound punctured Klein bottles

Abstract

We investigate the properties of knots in ${\mathbb{S}}^3$ which bound punctured Klein bottles, such that a pushoff of the knot has zero linking number with the knot, ie 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 punctured 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 $\mathbb{Z}\left[1∕2\right]$–homology ${\mathbb{B}}^4$ 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 $\left(2,p\right)$–cables of $K$ and $J$ are $\mathbb{Z}\left[1∕2\right]$–concordant if and only if $K$ and $J$ are $\mathbb{Z}\left[1∕2\right]$–concordant. In particular, if the $\left(2,1\right)$–cable of a knot $K$ is slice, $K$ is slice in a $\mathbb{Z}\left[1∕2\right]$–homology ball.