#### Volume 18, issue 3 (2014)

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Covering link calculus and the bipolar filtration of topologically slice links

### Jae Choon Cha and Mark Powell

Geometry & Topology 18 (2014) 1539–1579
##### Abstract

The bipolar filtration introduced by T Cochran, S Harvey and P Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice $1$–bipolar knots which are not $2$–bipolar. For knots, this is the highest known level at which the filtration does not stabilize. For the case of links with two or more components, we prove that the filtration does not stabilize at any level: for any $n$, there are topologically slice links which are $n$–bipolar but not $\left(n+1\right)$–bipolar. In the proof we describe an explicit geometric construction which raises the bipolar height of certain links exactly by one. We show this using the covering link calculus. Furthermore we discover that the bipolar filtration of the group of topologically slice string links modulo smooth concordance has a rich algebraic structure.

##### Keywords
covering link calculus, concordance, bipolar filtration
##### Mathematical Subject Classification 2010
Primary: 57M25, 57N70
##### Publication
Received: 1 May 2013
Accepted: 7 October 2013
Published: 7 July 2014
Proposed: Peter Teichner
Seconded: Robion Kirby, Yasha Eliashberg
##### Authors
 Jae Choon Cha Department of Mathematics Pohang University of Science and Technology Gyungbuk Pohang 790-784 South Korea and School of Mathematics Korea Institute for Advanced Study Seoul 130–722 South Korea Mark Powell Department of Mathematics Indiana University Rawles Hall 831 East 3rd Street Bloomington, IN 47405 USA