Covering link calculus and the bipolar filtration of topologically slice links

TL;DR

This paper investigates the bipolar filtration of topologically slice links, demonstrating that for links with multiple components, the filtration does not stabilize at any level, and introduces a geometric construction to analyze this behavior.

Contribution

It proves that the bipolar filtration for multi-component links does not stabilize at any level and introduces an explicit geometric method to increase bipolar height.

Findings

Filtration does not stabilize at any level for links with multiple components.

Explicit geometric construction raises bipolar height by one.

The bipolar filtration group has a rich algebraic structure.

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.