A construction of slice knots via annulus modifications

Abstract

We define an operation on homology ${B}^4$ which we call an $n$-twist annulus modification. We give a new construction of smoothly slice knots and exotically slice knots via $n$-twist annulus modifications. As an application, we present a new example of a smoothly slice knot with non-slice derivatives. Such examples were first discovered by Cochran and Davis. Also, we relate $n$-twist annulus modifications to $n$-fold annulus twists which was first introduced by Osoinach, then has been used by Abe and Tange to construct smoothly slice knots. Furthermore we consider $n$-twist annulus modifications in more general setting to show that any exotically slice knot can be obtained by the image of the unknot in the boundary of a smooth $4$-manifold homeomorphic to ${B}^4$ after an annulus modification.