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Description

The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in S^3 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 $\mathbb{R}P^1$) 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.