Algebraic & Geometric Topology

Slice knots which bound punctured Klein bottles

Arunima Ray

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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.

Article information

Algebr. Geom. Topol., Volume 13, Number 5 (2013), 2713-2731.

Received: 14 March 2013
Revised: 15 March 2013
Accepted: 17 March 2013
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

knot concordance


Ray, Arunima. Slice knots which bound punctured Klein bottles. Algebr. Geom. Topol. 13 (2013), no. 5, 2713--2731. doi:10.2140/agt.2013.13.2713.

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