Shake slice and shake concordant knots (Q2826649)

The 4-manifold \(W^r_K\) obtained by adding a 2-handle to the 4-ball along a knot \(K\) with framing \(r\) is homotopy equivalent to the 2-sphere. If \(K\) is a slice knot the homotopy equivalence is realized by a smooth embedding \(S^2\subset{W^r_K}\). The latter property is relativized here; two knots \(K_0\) and \(K_1\) are \textit{\(r\)-shake concordant} if the 4-manifold \(W^r_{K_0,K_1}=W^r_{K_0}\#W^r_{K_1}\) obtained by adding 2-handles to \(S^3\times[0,1]\) along \(K_0\times\{0\}\) and \(K_1\times\{1\}\) with framings \(r\) contains a smooth 2-sphere representing the sum of the homology classes determined by each 2-handle. (The focus here is on the smooth category, but there are parallel considerations for TOP locally flat 2-spheres and concordances.)NEWLINENEWLINEThe main result is a criterion for two knots to be \(r\)-shake concordant, in terms of ordinary concordance and satellite constructions. In particular, \(K\) is \(r\)-shake slice if and only if there is a winding number 1 pattern \(P\) such that \(P(U)\) is a ribbon knot (where \(U\) is the unknot) and the \(r\)-twisted satellite \(P_r(K)\) is a slice knot. The paper also gives the first examples of 0-shake concordant knots which are not concordant. This is not easy, since such knots must have the same algebraic concordance class. (There are even examples which are TOP slice!) A key tool is a lower bound for the genus of a smooth embedded surface representing a generator of \(H_2(W^r_K;\mathbb{Z})\), in terms of Thurston-Bennequin numbers and rotation numbers of Legendrian representatives of the knot.