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Topological properties of Fortuin-Kasteleyn clusters are studied on thetorus. Namely, the probability that their topology yields a given subgroup ofthe first homology group of the torus is computed for Q=1, 2, 3 and 4. Theexpressions generalize those obtained by Pinson for percolation (Q=1).Numerical results are also presented for three tori of different moduli. Theyagree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement isnot ruled out but what seems logarithmic corrections makes it harder to decide.Comment: 12 pages, 2 figures, submitted to J. Stat. Phy

Homology of Fortuin-Kasteleyn clusters of Potts models on the torus