### Property P for Some Classes of Knots

Rutgers University, 1986

Abstract

Let K be a knot in S^{3}. It is conjectured that all non-trivial knots in S^{3} have (homotopy) property P, i.e., no non-trivial Dehn surgery on a non-trivial knot in S^{3} yields a (homotopy) 3-sphere. There are many classes of knots known to have property P. We add two new classes to this collection. Suppose K is a genus one knot, with F a genus one Seifert surface for K. In the first chapter it is shown that if S^{3}-F is compressible, then K has property P. The proof relies on the Z_{2} Smith conjecture and elementary combinatorics. In the second chapter we show that if K is a non-trivial band-connect sum, then K has property P. As an immediate corollary we obtain a theorem, originally due to M. Scharlemann, which states that if
the band-connect sum of K_{1} and K_{2} is the unknot, then K_{1}, K_{2} and the connecting band are all trivial. The proof requires the Kirby calculus and the solution to the Poenaru conjecture (Gabai).