The twisting index in semitoric systems

TL;DR

This paper explores the twisting index invariant in semitoric integrable systems, providing new formulations and computing it for a complex family with multiple focus-focus points, advancing the classification of these systems.

Contribution

It introduces new dynamical, geometric, and topological interpretations of the twisting index and computes it for a novel family with multiple focus-focus singularities.

Findings

New formulations of the twisting index in terms of action differences, Taylor series, and homology.

First computation of the twisting index for systems with more than one focus-focus point.

Complete invariant set determination for a new family of semitoric systems.

Abstract

Semitoric integrable systems were symplectically classified by Pelayo and Vu Ngoc in 2009-2011 in terms of five invariants. Four of these invariants were already well-understood prior to the classification, but the fifth invariant, the so-called twisting index invariant, came as a surprise. Intuitively, the twisting index encodes how the structure in a neighborhood of a focus-focus fiber compares to the large-scale structure of the semitoric system and it was originally defined by comparing certain momentum maps. In the first half of the present paper, we produce several new formulations of the twisting index which give rise to dynamical, geometric, and topological interpretations. More specifically, we describe it in terms of differences of action variables, Taylor series, and homology cycles. In the second half of the paper, we compute the twisting index invariant of a specific family of systems with two focus-focus singular points (the so-called generalized coupled angular momenta), which is the first time that the twisting index has been computed for a system with more than one focus-focus point. Moreover, we also compute the terms of the Taylor series invariant up to second order. Since the other invariants of this family were already computed, this becomes the third family of semitoric systems for which all invariants are known, after the coupled spin oscillators and the coupled angular momenta.