$\mathbb{Z}_2$ topological invariants from the Green's function diagonal zeros

We investigate the relationship between the analytical properties of the Green's function and $\mathbb{Z}_2$ topological insulators, focusing on three-dimensional inversion-symmetric systems. We show that the diagonal zeros of the Green's function in the orbital basis provide a direct and visual way to calculate the strong and weak $\mathbb{Z}_2$ topological invariants. We introduce the surface of crossings of diagonal zeros in the Brillouin zone, and show that it separates time-reversal invariant momenta (TRIMs) of opposite parity in two-band models, enabling the visual computation of the $\mathbb{Z}_2$ invariants by counting the relevant TRIMs on either side. In three-band systems, a similar property holds in every case except when a trivial band is added in the band gap of a non-trivial two-band system, reminiscent of the band topology of fragile topological insulators.