A note on cabled slice knots and reducible surgeries (Q2363551)

It may happen that Dehn surgery on a knot in \(S^3\) results in a reducible \(3\)-manifold, that is, a \(3\)-manifold which can be split by a \(2\)-sphere into two non-trivial \(3\)-manifolds. The question of interest to the author here is whether those \(3\)-manifolds are themselves irreducible, i.e. can Dehn surgery on a knot in \(S^3\) result in a \(3\)-manifold which can be split by \(2\)-spheres into three non-trivial pieces? He shows this cannot happen if the knot is a slice knot. His proof utilizes the correction terms from Heegard Floer homology. He also gives a necessary condition in terms of the Heegard Floer homology for a positive cable to be slice.