Communications in Analysis and Geometry

Volume 12 (2004)

Number 2

Non-integral Toroidal Dehn Surgeries

Pages: 417 – 485



John Luecke

C. McA. Gordon


If we perform a non-trivial Dehn surgery on a hyperbolic knot in the 3- sphere, the result is usually a hyperbolic 3-manifold. However, there are exceptions: there are hyperbolic knots with surgeries that give lens spaces [1], small Seifert fiber spaces [2], [5], [7], [19], and toroidal manifolds, that is, manifolds containing (embedded) incompressible tori [6], [7]. In particular, Eudave-Muñoz [6] has explicitly described an infinite family of hyperbolic knots k(��, m, n, p), each of which has a specific half-integral toroidal surgery. (These are the only known examples of non-trivial, non-integral, non-hyperbolic surgeries on hyperbolic knots.) Here we show that these knots are the only hyperbolic knots with non-integral toroidal surgeries.

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