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

TL;DR

This paper constructs a smooth family of integrable systems on a 4D symplectic manifold, transitioning from a toric to a semitoric system with four focus-focus points, including a special case with double pinched tori.

Contribution

It introduces a new family of integrable systems with specific singularities, notably including systems with double pinched tori, expanding the understanding of semitoric systems.

Findings

Family transitions from toric to semitoric systems.

Presence of four focus-focus singularities in the family.

Explicit parametrization of a double pinched torus fiber.

Abstract

We construct a 1-parameter family $F_t=(J, H_t)_{0 \leq t \leq 1}$ 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.