Elementary surgery along a torus knot

Abstract

In this paper a classification of the manifolds obtained by a (p, q) surgery along an (r, s) torus knot is given. If |σ| = |rsp + q| ≠ 0, then the manifold is a Seifert manifold, singularly fibered by simple closed curves over the 2-sphere with singularities of types α1= s, α2 = r, and α3= |σ|. If M = 1, then there are only two singular fibers of types α1= s, α2 = r, and the manifold is a lens space L(|q|, ps2). If |σ| =0, then the manifold is not singularly fibered but is the connected sum of two lens spaces L(r, s)#L(s, r). It is also shown that the torus knots are the only knots whose comple¬ments can be singularly fibered. © 1971 Pacific Journal of Mathematics.