Covering link calculus and iterated Bing doubles

Cha, Jae Choon; Kim, Taehee

doi:10.2140/gt.2008.12.2173

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Covering link calculus and iterated Bing doubles

Jae Choon Cha and Taehee Kim

Geometry & Topology 12 (2008) 2173–2201

DOI: 10.2140/gt.2008.12.2173

Abstract

We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for $n > 1$ the $(n + 1)$ –st iterated Bing double of a knot is rationally slice if and only if the $n$ –th iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying a given link. We prove certain similar geometric obstructions for $n \leq 1$ as well. Our results are sharp enough to conclude, when combined with algebraic invariants, that if the $n$ –th iterated Bing double of a knot is slice for some $n$ , then the knot is algebraically slice. Also our geometric arguments applied to the smooth case show that the Ozsváth–Szabó and Manolescu–Owens invariants give obstructions to iterated Bing doubles being slice. These results generalize recent results of Harvey, Teichner, Cimasoni, Cha and Cha–Livingston–Ruberman. As another application, we give explicit examples of algebraically slice knots with nonslice iterated Bing doubles by considering von Neumann $ρ$ –invariants and rational knot concordance. Refined versions of such examples are given, that take into account the Cochran–Orr–Teichner filtration.

Keywords

iterated Bing doubles, covering links, slice links, rational concordance, von Neumann $\rho$–invariants, Heegaard Floer invariants

Mathematical Subject Classification 2000

Primary: 57M25, 57N70

References

Publication

Received: 8 January 2008

Revised: 23 June 2008

Accepted: 24 May 2008

Published: 8 August 2008

Proposed: Peter Teichner

Seconded: Ron Stern, Cameron Gordon

Authors

Jae Choon Cha
Department of Mathematics and Pohang Mathematics Institute Pohang University of Science and Technology Pohang, Gyungbuk 790–784 Republic of Korea

Taehee Kim
Department of Mathematics Konkuk University Seoul 143–701 Republic of Korea

Volume 12, issue 4 (2008)