Markov numbers and Lagrangian cell complexes in the complex projective plane

We study Lagrangian embeddings of a class of two-dimensional cell complexes L-p,L-q into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type (1/p(2)) (pq - 1, 1) (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into CP2 then p is a Markov number and we completely characterise q. We also show that a collection of Lagrangian pinwheels L-pi,L-qi, i = 1, ... , N, cannot be made disjoint unless N <= 3 and the p(i) form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a Q-Gorenstein smoothing whose general fibre is CP2.