#### Volume 13, issue 5 (2013)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Slice knots which bound punctured Klein bottles

### Arunima Ray

Algebraic & Geometric Topology 13 (2013) 2713–2731
##### 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 $ℤ\left[1∕2\right]$–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 $ℤ\left[1∕2\right]$–concordant if and only if $K$ and $J$ are $ℤ\left[1∕2\right]$–concordant. In particular, if the $\left(2,1\right)$–cable of a knot $K$ is slice, $K$ is slice in a $ℤ\left[1∕2\right]$–homology ball.

knot concordance
Primary: 57M25
##### Publication
Revised: 15 March 2013
Accepted: 17 March 2013
Published: 10 July 2013
##### Authors
 Arunima Ray Department of Mathematics Rice University MS-136, PO Box-1892 Houston, TX 77251-1892 USA http://math.rice.edu/~ar25/