  An analytic method to calculate the vortex number on a torus is constructed,
focusing on analytic vortex solutions to the Chern-Simons-Higgs theory, whose
governing equation is the so-called Jackiw-Pi equation. The equation is one of
the integrable vortex equations and is reduced to Liouville's equation. The
requirement of continuity of the Higgs field strongly restricts the
characteristics and the fundamental domain of the vortices. Also considered are
the decompactification limits of the vortices on a torus, in which "flux loss"
phenomena occasionally occur.
