Taylor series and twisting-index invariants of coupled spin-oscillators

TL;DR

This paper completes the classification of the coupled spin-oscillator system by computing higher order terms of the Taylor series invariant and the twisting index, revealing symmetry properties and superintegrability at zero energy.

Contribution

It introduces the calculation of higher order terms of the Taylor series invariant and the twisting index for the coupled spin-oscillator, advancing the classification of semitoric systems.

Findings

Higher order terms of the Taylor series invariant are computed.

The twisting index for the coupled spin-oscillator is determined.

Symmetry properties imply superintegrability at zero energy level.

Abstract

About six years ago, semitoric systems on 4-dimensional manifolds were classified by Pelayo & Vu Ngoc by means of five invariants. A standard example of such a system is the coupled spin-oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$. Calculations of three of the five semitoric invariants of this system (namely the number of focus-focus singularities, the generalised semitoric polygon, and the height invariant) already appeared in the literature, but the so-called twisting index was not yet computed and, of the so-called Taylor series invariant, only the linear terms were known. In the present paper, we complete the list of invariants for the coupled spin-oscillator by calculating higher order terms of the Taylor series invariant and by computing the twisting index. Moreover, we prove that the Taylor series invariant has certain symmetry properties that make the even powers in one of the variables vanish and allow us to show superintegrability of the coupled spin-oscillator on the zero energy level.