Vol. 53, No. 2, 1974

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

Louise Elizabeth Moser

Vol. 53 (1974), No. 2, 519–523

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
Received: 19 April 1974
Published: 1 August 1974
Louise Elizabeth Moser