From compact semi-toric systems to Hamiltonian S¹-spaces

We show how any labeled convex polygon associated to a compact semi-toric system, as defined by Vũ ngọc, determinesKarshon's labeled directed graph which classifies the underlyingHamiltonian S¹-space up to isomorphism. Then we characterizeadaptable compact semi-toric systems, i.e. those whoseunderlying Hamiltonian S¹-action can be extended to an effectiveHamiltonian T²-action, as those which have at least oneassociated convex polygon which satisfies the Delzant condition.