Volume 13, issue 5 (2013)

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Slice knots which bound punctured Klein bottles

Arunima Ray

Algebraic & Geometric Topology 13 (2013) 2713–2731

We investigate the properties of knots in S3 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 [12]–homology B4 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 [12]–concordant if and only if K and J are [12]–concordant. In particular, if the (2,1)–cable of a knot K is slice, K is slice in a [12]–homology ball.

knot concordance
Mathematical Subject Classification 2010
Primary: 57M25
Received: 14 March 2013
Revised: 15 March 2013
Accepted: 17 March 2013
Published: 10 July 2013
Arunima Ray
Department of Mathematics
Rice University
MS-136, PO Box-1892
Houston, TX 77251-1892