Agnes Scott College

Abigail Thompson

Property P for Some Classes of Knots
Rutgers University, 1986


Let K be a knot in S3. It is conjectured that all non-trivial knots in S3 have (homotopy) property P, i.e., no non-trivial Dehn surgery on a non-trivial knot in S3 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 S3-F is compressible, then K has property P. The proof relies on the Z2 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 K1 and K2 is the unknot, then K1, K2 and the connecting band are all trivial. The proof requires the Kirby calculus and the solution to the Poenaru conjecture (Gabai).