Elementary surgery along a torus knot

Louise Elizabeth Moser

Vol. 38 (1971), No. 3, 737–745

DOI: 10.2140/pjm.1971.38.737

Abstract

In this paper a classification of the manifolds obtained by a (p,q) surgery along an (r,s) lorus 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 α₈ = |σ|. If |σ| = 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 complements can be singularly fibered.

Mathematical Subject Classification

Primary: 55F55

Secondary: 57C45

Milestones

Received: 19 October 1970

Published: 1 September 1971