The Height Invariant of a Four-Parameter Semitoric System with Two Focus–Focus Singularities

Abstract

AbstractSemitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vũ Ngọc about a decade ago in terms of five symplectic invariants. If a semitoric system has several focus–focus singularities, then some of these invariants have multiple components, one for each focus–focus singularity. Their computation is not at all evident, especially in multi-parameter families. In this paper, we consider a four-parameter family of semitoric systems with two focus–focus singularities. In particular, apart from the polygon invariant, we compute the so-called height invariant. Moreover, we show that the two components of this invariant encode the symmetries of the system in an intricate way.