Unknotting number 21 knots are slice in K3

Abstract: A question in knot theory that has become very popular recently is to classify what knots bound a smooth disc in X − int(B4 ), where X is a given closed 4-manifold. We study the case when X is the K3 surface, and prove that every knot that can be unknotted with at most 21 crossing changes, bounds a smooth disc in K3 − int(B4 ). Our proof is constructive and based on the existence of a plumbing tree of 22 spheres in K3. We present a simple but flexible technique to simultaneously remove multiple singularities of immersed surfaces in 4-manifolds.