Geometry & Topology

Covering link calculus and the bipolar filtration of topologically slice links

Jae Choon Cha and Mark Powell

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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 (n+1)–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.

Article information

Geom. Topol., Volume 18, Number 3 (2014), 1539-1579.

Received: 1 May 2013
Accepted: 7 October 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57N70: Cobordism and concordance

covering link calculus concordance bipolar filtration


Cha, Jae Choon; Powell, Mark. Covering link calculus and the bipolar filtration of topologically slice links. Geom. Topol. 18 (2014), no. 3, 1539--1579. doi:10.2140/gt.2014.18.1539.

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