Trifectas for $T_N$ in 5d

The trinions $T_N$ are a class of 5d $\mathcal{N}=1$ superconformal field theories (SCFTs) realized as M-theory on $$\mathbb{C}$^3/$\mathbb{Z}$_N \times $\mathbb{Z}$_N$\mathrm{.}Weapplyto$T_N$, as well as closely-related SCFTs that are obtained by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view - toric and gluing of compact surfaces as well as combined fiber diagrams - to magnetic quivers and Hasse diagrams.