  Minimal tori that are linearly full in the 3-sphere possess a natural
invariant g called their spectral genus, which was introduced by Hitchin. We
show that for each g>0, there are countably many real g-dimensional families of
minimally immersed tori with spectral genus g (two of these dimensions are just
reparametrisations of the torus). Each family maps from a fixed torus of
rectangular conformal type.
