  In previous work by Coates, Galkin, and the authors, the notion of mutation
between lattice polytopes was introduced. Such a mutation gives rise to a
deformation between the corresponding toric varieties. In this paper we study
one-step mutations that correspond to deformations between weighted projective
planes, giving a complete characterisation of such mutations in terms of
T-singularities. We show also that the weights involved satisfy Diophantine
equations, generalising results of Hacking-Prokhorov.
