Manifolds with multiple knot-surgery descriptions

Every closed connected orientable three-manifold can be constructed by surgery along an appropriately chosen link in the three-sphere, S3 ((10), (15)). Such a link-surgery description is never unique, but the equivalence between different descriptions has been explicitly identified ((8),(3)). However, the situation with regard to manifolds that can be obtained by surgery along a knot in S3 is less clear ((6), p. 47): some manifolds are known to have at least two knot-surgery descriptions ((12), (14), p. 270, (1), (11)), whilst certain manifolds (e.g. S3 and S2 × S1) are widely suspected to have just one (see problems 1·15, 1·16 of (9)).