- 4607B South Hall
In this talk, we'll be concerned with producing reducible 3-manifolds via Dehn surgery on a knot in the three-sphere. The proposed classification of such surgeries is due to Gonzalez-Acuna and Short, known as the Cabling Conjecture, which asserts that only a certain surgery on cable and torus knots (and 0-surgery on the unknot) can give such manifolds. Using these surgeries as a guide, a large amount of progress toward the conjecture has been established which essentially tells you that an arbitrary reducible Dehn surgery is coarsely similar to the cabled reducible surgery. In a similar spirit, it should be the case that all reducible Dehn surgeries on nontrivial knots give precisely two irreducible connected summands, sometimes referred to as the Two Summands Conjecture. Using the main combinatorial object appearing first in the proof of the Knot Complement Problem due to Gordon and Luecke, we are able to restrict any surgery coefficient producing more than two summands to being less than or equal to the bridge number of the purported knot. A consequence of this is the completion of the Two Summands Conjecture for positive braid closures and knots with bridge number less than or equal to five, and another proof of the Cabling Conjecture for knots with bridge number less than or equal to 3.