SU(2)-cyclic surgeries and the pillowcase

We study knots in S3 with infinitely many SU(2)-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into SU(2) has cyclic image. We show that for every such nontrivial knot K, its set of SU(2)-cyclic slopes is bounded and has a unique limit point, which is both a rational number and a boundary slope for K. We also show that such knots are prime and have infinitely many instanton L‑space surgeries. Our methods include the application of holonomy perturbation techniques to instanton knot homology, using a strengthening of recent work by the second author