A family of semitoric systems with four focus-focus singularities and two double pinched tori

Abstract

We construct a 1-parameter family $F_t=(J,H_t{)}_{0\le t\le 1}$ of integrable systems on a compact $4$-dimensional symplectic manifold $(M,\omega )$ that changes smoothly from a toric system $F_0$ with eight elliptic-elliptic singular points via toric type systems to a semitoric system $F_t$ for $t^{-}<t<t^{+}$. These semitoric systems $F_t$ have precisely four elliptic-elliptic and four focus-focus singular points. Moreover, at $t=\frac{1}{2}$, the system has precisely two focus-focus fibres each of which contains exactly two focus-focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.

arXiv comment50 pages, 15 figures; some references added, list of references updated, 1 paragraph in introduction added