Covering Link Calculus and Iterated Bing Doubles

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 nth 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 ≤ 1 as well. Our results are sharp enough to conclude, when combined with algebraic invariants, that if the nth iterated Bing double of a knot is slice for some n, then the knot is algebraically slice. This generalizes results of Cha and Cha-Livingston-Ruberman. As another application, we give explicit examples of algebraically slice knots with non-slice iterated Bing doubles by considering von Neumann ρ-invariants and rational knot concordance. We also give more such examples by taking into account Cochran-Orr-Teichner’s filtrations on knots and links.