Title: Taylor series and twisting-index invariants of coupled spin-oscillators,
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.