SU(2)-cyclic surgeries and the pillowcase

A 3-manifold is called SU(2)-cyclic if every homomorphism from its
fundamental group to SU(2) has cyclic image.  In this talk, we will
study the question of when knots in S^3 have infinitely many
SU(2)-cyclic surgeries.  We will show that for any such knot, the set
of SU(2)-cyclic surgery slopes has a single limit point, and then by
applying holonomy perturbations to Kronheimer and Mrowka’s instanton
knot homology, we will show that this limit point is finite.  We thus
show that for any nontrivial knot, the set of slopes of SU(2)-cyclic
surgeries is bounded.  If time permits, we will also use some more
holonomy perturbations to show that the only two-bridge knots with
infinitely many SU(2)-cyclic surgeries are the (2,2n+1) torus knots.
This is joint work with Raphael Zentner.