The height invariant of a four-parameter semitoric system with two focus-focus singularities

TL;DR

This paper computes the height invariant for a four-parameter semitoric system with two focus-focus singularities, revealing how the invariant's components encode system symmetries, advancing understanding of multi-parameter integrable systems.

Contribution

It provides the first explicit computation of the height invariant in a multi-parameter semitoric system with multiple focus-focus singularities.

Findings

Computed the height invariant for a specific four-parameter family

Revealed how invariant components encode system symmetries

Enhanced understanding of invariants in multi-focus-focus semitoric systems

Abstract

Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vu Ngoc 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.