Covering link calculus and iterated Bing doubles

TL;DR

This paper introduces a new geometric obstruction to the sliceness of iterated Bing doubles of knots, linking it to algebraic invariants and providing explicit examples and applications in knot theory.

Contribution

It establishes a geometric obstruction criterion for iterated Bing doubles being slice, connecting it with algebraic sliceness and invariants like von Neumann rho-invariants.

Findings

Iterated Bing doubles are rationally slice iff their previous iteration is rationally slice.

Algebraically slice knots can have non-slice iterated Bing doubles.

Ozsvath-Szabo and Manolescu-Owens invariants obstruct sliceness of 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 (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 <= 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 Ozsvath-Szabo 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 non-slice 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.