A note on the symplectic classification of almost-toric systems

Abstract

Since simple semitoric systems were classified about fifteen years ago, and semitoric systems five years ago, we want to move a step forward to almost-toric systems. We give a classification of compact almost-toric systems in dimension four up to fiber-preserving symplectomorphisms, in terms of the base, Taylor series, and twisting indices, analogous to the five invariants for semitoric systems. For convenience, we specify an ordering of focus-focus values and a choice of two cut rays at each of them.