Semitoric Families on Pentagon Spaces

Semitoric systems are a special type of 4-dimensional integrable system where one of the functions is the moment map of a Hamiltonian $S^1$-action. While their classification is well understood thanks to the work of Pelayo and V{\~u} Ng\d{o}c, relatively few explicit examples are known. Recently, Le Floch and Palmer introduced semitoric transition families in which a singular point transitions between elliptic-elliptic type and focus-focus type as the parameter varies. In this paper, we construct new semitoric transition families on pentagon spaces by interpolating two semitoric systems of toric type. More precisely, we exhibit semitoric transition families, each with at least one transition point, on pentagon spaces of different diffeotypes, all defined by the same explicit interpolation formula.