Small Heegaard genus and SU(2)

Let Y be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if H 1 .Y I Z/ has order 1, 3, or 5, then there is a representation 1 .Y / !SU.2/ with nonabelian image.Similarly, if H 1 .Y I Z/ has order 2 then we find a nonabelian representation 1 .Y / !SO.3/.We also prove that a knot K in S 3 is a trefoil if and only if there is a unique conjugacy class of irreducible representations 1 .S 3 n K/ !SU.2/ sending a fixed meridian to diag.i; i /.