  Constant mean curvature (CMC) tori in Euclidean 3-space are described by an
algebraic curve, called the spectral curve, together with a line bundle on this
curve and a point on $ S ^ 1 $, called the Sym point. For a given spectral
curve the possible choices of line bundle and Sym point are easily described.
The space of spectral curves of tori is totally disconnected. Hence to
characterise the "moduli space" of CMC tori one should, for each genus $g$,
determine the closure $\overline{\mathcal{P}^g}$ of spectral curves of CMC tori
within the spectral curves of CMC planes having spectral genus $g$. We identify
a real subvariety $\mathcal{R}^g$ and a subset
$\mathcal{S}^g\subseteq\mathcal{R}^g $ such that
$\mathcal{R}^g_{\text{max}}\subseteq\overline{\mathcal{P}^g}\subseteq\mathcal{S}^g$,
where $\mathcal{R}^g_{\text{max}}$ denotes the points of $\mathcal{R}^g$ having
maximal dimension. The lowest spectral genus for which tori exist is $g=2$ and
in this case
$\mathcal{R}^2=\mathcal{R}^2_{\text{max}}=\overline{\mathcal{P}^2}=\mathcal{S}^2$.
For $g>2 $, we conjecture that
$\mathcal{R}^g\supsetneq\mathcal{R}^g_{\text{max}}=\mathcal{S}^g$. We give a
number of alternative characterisations of $\mathcal{R}^g_{\text{max}}$ and in
particular introduce a new integer invariant of a CMC plane of finite type,
called its winding number.
