Steven Sivek (Imperial College, London) will speak in the geometry seminar on the 6th of March. The talk will take place in the Salle de Profs (9th floor, NO) at 13.30pm. Steven’ title is “SU(2)-cyclic surgeries and the pillowcase” and the abstract is below.
The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in the 3-sphere other than a torus knot has at most two nontrivial cyclic surgeries. In this talk, we investigate the weaker notion of SU(2)-cyclic surgeries on a knot, meaning surgeries whose fundamental groups only admit SU(2) representations with cyclic image. By studying the image of the SU(2) character variety of a knot in the “pillowcase”, we will show that if it has infinitely many SU(2)-cyclic surgeries, then the corresponding slopes (viewed as a subset of RP^1) have a unique limit point, which is a finite, rational number, and that this limit is a boundary slope for the knot. As a corollary, it follows that for any nontrivial knot, the set of SU(2)-cyclic surgery slopes is bounded. This is joint work with Raphael Zentner.