On the impossibility of obtaining S² × S¹ by elementary surgery along a knot

Louise Elizabeth Moser

Vol. 53 (1974), No. 2, 519–523

DOI: 10.2140/pjm.1974.53.519

Abstract

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.

Mathematical Subject Classification

Primary: 55A25

Secondary: 57A10

Milestones

Received: 19 April 1974

Published: 1 August 1974