Yohann Le Floch, Universite de Strasbourg et CNRS, France, and Joseph Palmer, University of Illinois at Urbana-Champaign, Illinois, United States, and University of Antwerp, Antwerpen, Belgium

• 1. Introduction2. Preliminaries on integrable and semitoric systems3. Fixed-$\mathbb {S}^1$ families4. Toric type blowups and blowdowns5. Semitoric transition families on Hirzebruch surfaces6. Explicit semitoric families on $W_{1}(\alpha ,\beta )$7. Explicit semitoric families on $W_{2}(\alpha ,\beta )$A. Comparison of the systems in and using the height invariant