  A Fortuin-Kasteleyn cluster on a torus is said to be of type $\{a,b\},
a,b\in\mathbb Z$, if it possible to draw a curve belonging to the cluster that
winds $a$ times around the first cycle of the torus as it winds $-b$ times
around the second. Even though the $Q$-Potts models make sense only for $Q$
integers, they can be included into a family of models parametrized by
$\beta=\sqrt{Q}$ for which the Fortuin-Kasteleyn clusters can be defined for
any real $\beta\in (0,2]$. For this family, we study the probability
$\pi({\{a,b\}})$ of a given type of clusters as a function of the torus modular
parameter $\tau=\tau_r+i\tau_i$. We compute the asymptotic behavior of some of
these probabilities as the torus becomes infinitely thin. For example, the
behavior of $\pi(\{1,0\})$ is studied along the line $\tau_r=0$ and
$\tau_i\to\infty$. Exponents describing these behaviors are defined and related
to weights $h_{r,s}$ of the extended Kac table for $r,s$ integers, but also
half-integers. Numerical simulations are also presented. Possible relationship
with recent works and conformal loop ensembles is discussed.
