## Geometry & Topology

### Covering link calculus and the bipolar filtration of topologically slice links

#### 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 $(n+1)$–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

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

Dates
Accepted: 7 October 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732801

Digital Object Identifier
doi:10.2140/gt.2014.18.1539

Mathematical Reviews number (MathSciNet)
MR3228458

Zentralblatt MATH identifier
1304.57012

#### Citation

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. https://projecteuclid.org/euclid.gt/1513732801

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