Manifolds obtained by surgery on an infinite number of knots in S3

Abstract

The construction of 3-manifolds via Dehn surgery on links in $S^3$ is an important technique in the classification of 3-manifolds. This paper describes a method of constructing infinite collections of distinct hyperbolic knots in $S^3$ which admit a longitudinal surgery yielding the same manifold. In one case, the knots constructed each admit a longitudinal surgery yielding the same hyperbolic manifold; in another case, the knots admit a longitudinal surgery yielding the same toroidal manifold. This answers a question formulated by Kirby in the Kirby problem list [R. Kirby (Ed.), Problems in low-dimensional topology, in: Geometric Topology, American Mathematical Society/International Press, 1997] in the affirmative, which asks if there is a homology 3-sphere, or any 3-manifold, that can be obtained by $n$ surgery on an infinite number of distinct knots.