  We study 3d $\mathcal{N}=2$ $U(1)$ Chern-Simons-matter QFT on a cylinder
$C\times\mathbb{R}$. The topology of $C$ gives rise to BPS sectors of
low-energy solitons known as kinky vortices, which interpolate between
(possibly) different vacua at the ends of the cylinder and at the same time
carry magnetic flux.
  We compute the spectrum of BPS vortices on the cylinder in an isolated Higgs
vacuum, through the framework of \emph{warped} exponential networks, which we
introduce. We then conjecture a relation between these and standard vortices on
$\mathbb{R}^2$, which are related to genus-zero open Gromov-Witten invariants
of toric branes. More specifically, we show that in the limit of large
Fayet-Iliopoulos coupling, the spectrum of kinky vortices on $C$ undergoes an
infinite sequence of wall-crossing transitions, and eventually stabilizes. We
then propose an exact relation between a generating series of stabilized CFIV
indices and the Gromov-Witten disk potential, and discuss its consequences for
the structure of moduli spaces of vortices.
