Dehn Surgery on Knots in 3-manifolds

It has been known for over 30 years that every closed connected orientable 3manifold is obtained by surgery on a link in S [8]. However, a classification of such 3-manifolds in terms of this surgery construction has remained elusive. This is due primarily to the lack of uniqueness of the surgery description. In [5], Kirby gave us a calculus of surgery diagrams. However, the lack of a ‘canonical’ surgical method of getting from one 3-manifold to another has hampered further progress. In this paper, we show that if one restricts attention to the case where a surgery curve is homotopically trivial in a 3-manifold, then one has the following uniqueness theorem. Theorem 4.1. For i = 1 and 2, let V i be a compact connected oriented 3-manifold, possibly with boundary, such that H1(V ; Q) 6= 0. Let Ki be a homotopically trivial knot in V , such that V i − Ki is irreducible and atoroidal. Let V i Ki(pi/qi) be the manifold obtained from V i by pi/qi Dehn surgery along Ki, where pi and qi are coprime integers. Then, there is a natural number C(V , K1), which depends only on V 1 and K1, with the following property. If |q2| > C(V , K1), then