The combinatorics of reducible Dehn surgeries

We study reducible Dehn surgeries on nontrivial knots in S³. The conjectured classification of such surgeries is known as the Cabling Conjecture, and partial progress toward the conjecture often comes in the form of a statement that an arbitrary reducible surgery resembles a cabled reducible surgery. One such resemblance is the Two Summands Conjecture: Dehn surgery on a knot in S³ can only produce a manifold with at most two irreducible connected summands. In the event that a reducible surgery on a knot K in S³ of slope r produces a manifold with more than two such summands, we show that |r| ≤ b, where b denotes the bridge number of K. As a consequence, we rule out this possibility for knots with b ≤ 5 and for positive braid closures. We also study reducible Dehn surgeries without the assumption that the reducible manifold contains more than two connected summands. Specifically, if P is an essential planar surface in the exterior of a hyperbolic knot which completes to a reducing sphere in this surgery, then it is shown that the number of boundary components of P is at least ten.