Covering link calculus and iterated Bing doubles

Abstract

We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for $n>1$ the $\left(n+1\right)$–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 $n\le 1$ 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 $\rho$–invariants and rational knot concordance. Refined versions of such examples are given, that take into account the Cochran–Orr–Teichner filtration.