Slice knots which bound punctured Klein bottles

We investigate the properties of knots in $S^{3}$ 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 $ℤ [1 ∕ 2]$ –homology $B^{4}$ 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 $(2, p)$ –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.

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Keywords

knot concordance

Mathematical Subject Classification 2010

Primary: 57M25

References

Publication

Received: 14 March 2013

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/

Volume 13, issue 5 (2013)

Arunima Ray

Abstract