Unknotting number 21 knots are slice in K3

Abstract

We prove that all knots with unknotting number at most 21 are smoothly slice in the K3 surface. We also prove a more general statement for 4-manifolds that contain a plumbing tree of spheres. Our strategy is based on a flexible method to remove double points of immersed surfaces in 4-manifolds by tubing over neighbourhoods of embedded trees. As a byproduct, we recover a classical result of Norman and Suzuki that every knot is smoothly slice in $S^2\times S^2$ and in ${\mathbb{CP}}^2\#\overline{{\mathbb{CP}}^2}$.

13 pages, 7 figures. Final version accepted for publication