Every knot with unknotting number \leq 21 bounds a disc in a punctured K3 surface

A question in knot theory that has become very popular recently is to classify what knots bound smooth discs in X - int(B^4), where X is a given closed 4-manifold. If X is a K3 surface, we show that every knot that can be unknotted with at most 21 crossing changes bounds a smooth disc in X - int(B^4). Our proof is constructive and based on the existence of a plumbing tree of 22 spheres in K3. We also prove a more general result when X is an elliptic surface E(n). This is joint work with Stefan Mihajlović.