A note on the symplectic classification of almost-toric systems

TL;DR

This paper advances the classification of almost-toric systems in four-dimensional symplectic geometry by extending invariants used in semitoric systems, providing a detailed framework for their symplectic classification.

Contribution

It introduces a classification scheme for compact almost-toric systems in four dimensions, generalizing the invariants used for semitoric systems.

Findings

Classification based on base, Taylor series, and twisting indices

Specification of focus-focus value ordering and cut rays

Extension of semitoric invariants to 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.