Elementary surgery along a
knot has been used in an attempt to construct a counterexample to the Poincaré
Conjecture. Certain classes of knots have been examined, but no counterexample has
yet been found. Another, and perhaps as interesting a question, is whether S2× S1
can be obtained by elementary surgery along a knot. In this paper the question is
answered in the negative for knots with nontrivial Alexander polynomial,
for composite knots, and for a large class of knots with trivial Alexander
polynomial—the simply doubled knots.