  We study knots in $S^3$ 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.
