Embedded spheres in the K3 surface Using the Seiberg-Witten theory, we know that every (smoothly) embedded $S^{2}$ in $K3$ with trivial normal bundle is null-homologous. We know we have a lot of interesting knotted $S^{2}$ inside $S^{4}$ but is it the only way we obtain such spheres in $K3$? I.e. do we know any embedded $S^{2}$ inside $K3$ with trivial normal bundle but is not contained in any 4-ball?      
 A: As far as I know, this is an open question. It seems quite reasonable that such a surface should exist, but off the top of my head I'm not quite sure how you would construct something like that. In general, finding spheres in 4-manifolds is tricky business. For example, if the complement of your sphere had cyclic fundamental group, then it would at least topologically lie inside some 4-ball. This doesn't necessarily imply that they smoothly lie inside a ball, however. In http://arxiv.org/abs/1310.1838 you'll find tori in K3 that topologically line inside a ball, but presumably don't have that property smoothly (concretely, these tori bound topologically embedded solid handlebodies in K3, but not smoothly embedded handlebodies).
Here is an equivalent problem to the problem that you pose: try to find a 4-manifold that has the same intersection form as K3, and whose fundamental group is normally generated by a single curve. Then surgery along that curve will give a null-homologous surface in (a potentially exotic) K3, and as long as the original 4-manifold  was not a connect sum with K3 to begin with, this surface won't lie in a ball.
