Contact structures and reducible surgeries

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus$g$must have slope$2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

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

The Cabling conjecture states that Dehn surgery on a nontrivial knot K in S3 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 S3 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.

PARALLELISM OF TWO STRINGS IN ALTERNATING TANGLES

We consider the parallelism of two strings in alternating tangles. We show that if there is a pair of parallel strings in an alternating tangle then its alternating diagrams satify certain conditions. As a corollary, for a knot admitting a decomposition into two alternating tangles with two or three strings, we prove that its non-trivial Dehn surgery yields a 3-manifold with an essential lamination. Hence such a knot has property P and satisfy the cabling conjecture.

Symmetric knots satisfy the cabling conjecture

The cabling conjecture states that a non-trivial knot K in the 3-sphere is a cable knot or a torus knot if some Dehn surgery on K yields a reducible 3-manifold. We prove that symmetric knots satisfy this conjecture. (Gordon and Luecke also prove this independently ([GLu3]), by a method different from ours.)