Narrow Results

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(α) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a central role.

It is shown that a hyperbolic knot in the 3-sphere admits at most nine integral surgeries yielding non-hyperbolike 3-manifolds; namely, 3-manifolds which are reducible or whose fundamental groups are not infinite word-hyperbolic.

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space. Like torus knots and Berge knots, many known L-space knots admit a Seifert fibered L-space surgery. We give a concrete example of a hyperbolic L-space knot which has no exceptional surgeries, in particular, no Seifert fibered surgeries.