Unique Surgery Descriptions along Knots

We prove that for any non-trivial knot K, infinitely many r-surgeries K(r) 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(r). 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.