  Recent advances in topological mechanics have revealed unusual phenomena such
as topologically protected floppy modes and states of self-stress that are
exponentially localized at boundaries and interfaces of mechanical networks. In
this paper, we explore the topological mechanics of epithelial tissues, where
the appearance of these boundary and interface modes could lead to localized
soft or stressed spots and play a role in morphogenesis. We consider both a
simple vertex model (VM) governed by an effective elastic energy and its
generalization to an active tension network (ATN) which incorporates active
adaptation of the cytoskeleton. By analyzing spatially periodic lattices at the
Maxwell point of mechanical instability, we find topologically polarized phases
with exponential localization of floppy modes and states of self-stress in the
ATN when cells are allowed to become concave, but not in the VM.
