No abstract

No abstract

Two (smooth or PL) knots K, K' in S 3 are equivalent if there exists a homeomorphism h: S 3 -• S 3 such that h(K) = K'. This implies that their complements S 3 -K and S 3 -K' are homeomorphic. Here we announce the converse implication.

Let M be a compact, connected, orientable, irreducible 3-manifold such that dM is a torus. An isotopy class c of unoriented simple closed curves in dM will be called a slope. A closed 3-manifold M(c) may be constructed by attaching a solid torus / to M so that c bounds a disk in J.If c and d are two slopes, we denote their (minimal) geometric intersection number by A(c, d). This result is sharp; Fintushel-Stern and Berge have given examples of hyperbolic knots in S 3 for which two Dehn surgeries give lens spaces. In the statements of the following corollaries we use rational numbers as in [R] to parametrize the nontrivial Dehn surgeries on a knot K in S 3 . We will denote by K(r) the result of r-surgery on K.

No abstract