  Recent experimental observations have suggested that topological defects can
facilitate the creation of sharp features in developing embryos. Whereas these
observations echo established knowledge about the interplay between geometry
and topology in two-dimensional passive liquid crystals, the role of activity
has mostly remained unexplored. In this article we focus on deformable shells
consisting of either polar or nematic active liquid crystals and demonstrate
that activity renders the mechanical coupling between defects and curvature
much more involved and versatile than previously thought. Using a combination
of linear stability analysis and three-dimensional computational fluid
dynamics, we demonstrate that such a coupling can in fact be tuned, depending
on the type of liquid crystal order, the specific structure of the defect (i.e.
asters or vortices) and the nature of the active forces. In polar systems, this
can drive a spectacular transition from spherical to toroidal topology, in the
presence of large extensile activity. Our analysis strengthens the idea that
defects could serve as topological morphogens and provides a number of
predictions that could be tested in in vitro studies, for instance in the
context of organoids.
