We study Lagrangian embeddings of a class of two-dimensional cell complexes 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 (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into then is a Markov number and we completely characterise . We also show that a collection of Lagrangian pinwheels , , cannot be made disjoint unless and the 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 –Gorenstein smoothing whose general fibre is .
"Markov numbers and Lagrangian cell complexes in the complex projective plane." Geom. Topol. 22 (2) 1143 - 1180, 2018. https://doi.org/10.2140/gt.2018.22.1143