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

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### 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