On the impossibility of obtaining<i>S</i>²×<i>S</i>¹by elementary surgery along a knot

Abstract

Elementary surgery along a knot has been used in an attempt to construct a counterexample to the Poincar Conjecture. Certain classes of knots have been examined, but no counterexample has yet been found. Another, and perhaps as interesting a question, is whether S 2 X S 1 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.