THERE ARE NO STRICT GREAT x-CYCLES AFTER A REDUCING OR P² SURGERY ON A KNOT

Abstract

The Cabling conjecture states that Dehn surgery on a nontrivial knot K in S 3 yields a reducible manifold only when K is a nontrivial cabled knot. One idea is to attack this problem with the techniques used by Gordon and Luecke in the Knot Complement Problem. This involves a combinatorial analysis of two intersecting planar graphs. In the context of reducible surgery, one of the two planar graphs will necessarily contain a Scharlemann cycle. So, we define a strict x-cycle to be any x-cycle which is not a Scharlemann cycle; likewise, for strict great x-cycles. We show that if the reducing sphere meets the core of the Dehn filling minimally, then strict great x-cycles are not permitted. Thus, strict great x-cycles can play a role similar to that of the Scharlemann cycle in the Knot Complement Problem. The obstruction of finding strict great x-cycles is considered an essential step in the program. A second conjecture states that Dehn surgery on a nontrivial knot K in S 3 yields a manifold containing an embedded projective plane only when K is a nontrivial cabled knot. We show how the Gordon and Luecke technique can be applied towards this conjecture by considering the spherical boundary of a regular neighborhood of the projective plane. And if the projective plane is chosen to meet the core of the Dehn filling minimally, we show that strict great x-cycles are not permitted.