Shake slice and shake concordant knots

Abstract

A crucial step in the surgery-theoretic program to classify smooth manifolds is that of representing a middle--dimensional homology class by a smoothly embedded sphere. This step fails even for the simple 4-manifolds obtained from the 4-ball by adding a 2-handle with framing r along some knot K in S^3. An r-shake slice knot is one for which a generator of the second homology of this 4-manifold can be represented by a smoothly embedded 2-sphere. It is not known whether there exist 0-shake slice knots that are not slice. We define a relative notion of shake sliceness of knots, which we call shake concordance, which is easily seen to be a generalization of classical concordance, and we give the first examples of knots that are 0-shake concordant but not concordant; these may be chosen to be topologically slice. Additionally, for each r we completely characterize r-shake slice and r-shake concordant knots in terms of concordance and satellite operators. Our characterization allows us to construct new families of possible r-shake slice knots that are not slice.