Annulus twist and diffeomorphic 4-manifolds (Q2845620)

The authors start with the question: Is there a 3-manifold which can be obtained by \(n\)-surgery on infinitely many mutually distinct knots. Several examples for \(n=4m, m\in\mathbb{Z}\) answer the above question, see for example [\textit{T. Saito} and \textit{M. Teragaito}, Pac. J. Math. 244, No. 1, 169--192 (2010; Zbl 1305.57019)]. The authors address a related question which can be phrased as follows: For a knot \(K\) in the 3-sphere \(S^3=\partial D^4\), denote by \(M_K(n)\) the 3-manifold obtained by \(n\)-surgery on \(K\) and by \(X_K(n)\) the smooth 4-manifold obtained from the 4-ball \(D^4\) by attaching a 2-handle along \(K\) with framing \(n\). Note that \(\partial X_K(n)\) is diffeomorphic to \(M_K(n)\). As a 4-dimensional analog of the initial question one can ask: Let \(n\) be an integer. Find infinitely many mutually distinct knots \(K_i\) such that \(X_{K_i}(n)\) is diffeomorphic to \(X_{K_j}(n)\) for each \(i,j\in \mathbb{N}\)?NEWLINENEWLINESome of the authors have given a positive answer to this question, for example see the two masters theses of M. Takeuchi and Y. Omae or a paper by \textit{M. Teragaito} [Int. Math. Res. Not. 2007, No. 9, Article ID rnm028, 16 p. (2007; Zbl 1138.57012)]. In the paper reviewed here the authors develop a framework to construct such examples for \(n=0\) and \(n=\pm 4\). The authors also study the \(n\)-shake genus and its relationship to the 4-ball genus of a knot. Finally they give a construction of homotopy 4-spheres from a slice knot with unknotting number one.