To each coprime pair of natural numbers is associated a rational homology 4-ball B_{p,q}; these are of interest in algebraic geometry and in constructions of smooth 4-manifolds. Evans and Smith have completely determined which of these may be embedded symplectically in CP^2; the answer coincides with an algebraic geometric result of Hacking and Prokhorov, and is described in terms of solutions to the Markov diophantine equation.
Using double branched covers, we exhibit an infinite family of such balls which embed smoothly but not symplectically in CP^2. We also describe an obstruction using Donaldson’s diagonalisation which may be used to show that no two of our examples may be embedded disjointly.