Slicing mixed Bing-Whitehead doubles

TL;DR

This paper investigates the sliceness properties of certain Whitehead doubles of Bing doubles, showing they are topologically but not smoothly slice under specific conditions, advancing understanding in knot concordance and 4-manifold topology.

Contribution

It demonstrates that Whitehead doubles of Bing doubles are topologically slice but not smoothly slice for knots with positive tau invariant, and extends results to links like the Hopf link.

Findings

Whitehead doubles of Bing doubles are topologically slice for certain knots.

These doubles are not smoothly slice, indicating a distinction between topological and smooth categories.

Results apply to links like the Hopf link and Borromean rings.

Abstract

We show that if K is any knot whose Ozsvath-Szabo concordance invariant tau(K) is positive, the all-positive Whitehead double of any iterated Bing double of K is topologically but not smoothly slice. We also show that the all-positive Whitehead double of any iterated Bing double of the Hopf link (e.g., the all-positive Whitehead double of the Borromean rings) is not smoothly slice; it is not known whether these links are topologically slice.