Unique Surgery Descriptions along Knots

We prove that for any non-trivial knot K, infinitely many r-surgeries K® along K have a unique surgery description along a knot. Moreover, we show that for any hyperbolic L-space knot K and infinitely many integer slopes n, the manifold K(n) has a unique surgery description. Here we say a 3-manifold M has a unique surgery description along a knot in S^3 if there is a unique pair (K,r) of a knot K and a slope r such that M is orientation-preservingly diffeomorphic to K®. This generalises the notion of characterising slopes. Conversely, we provide new families of manifolds with several distinct surgery descriptions along knots. More precisely, we construct for every non-zero integer m a knot K_m such that for any integer n, the manifold K_m(m+1/n) can also be obtained by surgery on another knot.