  We prove that for any integer $n$ there exist infinitely many different knots
in $S^3$ 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.
