Orateur: Nicholas Zufelt (Imperial College,London)
Titre: The combinatorics of reducible Dehn surgeries.

Résumé: The proposed classification of reducible Dehn surgeries on knots in the three-sphere is known as the Cabling Conjecture. A large amount of progress toward the conjecture has been established which forces an arbitrary reducible surgery to coarsely resemble 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.