Covering link calculus and the bipolar filtration of topologically slice links

Cha, Jae Choon¹ ; Powell, Mark²

¹ 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
² Department of Mathematics, Indiana University, Rawles Hall, 831 East 3rd Street, Bloomington, IN 47405, USA

Geometry & topology, Tome 18 (2014) no. 3, pp. 1539-1579.

Résumé

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.

Détail

DOI : 10.2140/gt.2014.18.1539

Classification : 57M25, 57N70
Keywords: covering link calculus, concordance, bipolar filtration

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