Producing reducible 3-manifolds by surgery on a knot

Abstract

IT HAS long been conjectured that surgery on a knot in S3 yields a reducible 3-manifold if and only if the knot is cabled, with the cabling annulus part of the reducing sphere (cf. [7.8, 9, 10, 111). One may regard the Poenaru conjecture (solved in [S]) as a special case of the above. More generally, one can ask when surgery on a knot in an arbitary 3-manifold A4 produces a reducible 3-manifold M'. But this problem is too complex, since, dually, it asks which knots in which manifolds arise from surgery on reducible 3-manifolds. In this paper we are able to show, approximately, that if M itself either contains a summand not a rational homology sphere or is a-reducible, and M' is reducible, then k must have been cabled and the surgery is via the slope of the cabling annulus. Thus the result stops short of proving the conjecture for M = S3, but (see below) does suffice to prove the conjecture for satellite knots. The results here are broader than this; for a context recall the main result of [3]: GABAI'S THEOREM. Let k be a knot in M = D2 x S' with nonzero wrapping number. If_ci' is a manifold obtained by non-trivial surgery on k then one of the following must hold: (1) M = D2 x S' = M' and both k and k' are 0 or l-bridge braids. (2) M' = WI # W,, where W2 is a closed 3-manifold and H, (W,) isjnite and non-tritial. (3) M' is irreducible and aM ' is incompressible.

The classification of Dehn fillings on the outer torus of a 1-bridge braid exterior which produce solid tori

“…We remark that in general M (γ) may not be a solid torus for any γ since we are performing Dehn filling on the outer torus. Because of that, the powerful Reducible Surgery Theorem of Scharlemann [Sch,Theorem 6.1] does not apply to this situation.…”

Can Dehn surgery yield three connected summands?

“…The Cabling Conjecture is known to hold in many special cases [2], [8], [9], [12], [13], [17], [19].…”

Essential laminations in surgered 3-manifolds

“…We first assume that X is compact. If dX is incompressible in X-K, the result is part of [8,Theorem 6.1]. In general, choose B = (JB¡ to be the union of some disjoint compressing disks of dX in X -K, so that after cutting X along B, the new manifold Xx = X -lnt(N(B)) has boundary incompressible in Xx -K. By the hypothesis, dX has a compressing disk D such that dD does not bound a disk in X -K. Among all such compressing disks, we choose one, say Dx, that is transverse to B and minimizes \DX nB\.…”