Infinitely Many Knots Admitting the Same Integer Surgery and a Four-Dimensional Extension

We prove that for any integer |n| there exist infinitely many different knots in |S³| such that |n|-surgery on those knots yields the same |$3$|-manifold. In particular, when |$|n|=1$| homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, the first by a method of twisting along an annulus and the second by a generalization of this procedure. The latter family also solves a stronger version of Problem 3.6(D), that for any integer |n|⁠, there exist infinitely many mutually distinct knots such that 2-handle addition along each with framing |n| yields the same |$4$|-manifold.