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Following the suggestion of hep-th/0506249 and hep-th/0612011, we representquarter BPS dyons in N=4 supersymmetric string theories as string networkconfiguration and explore the role of genus two surfaces in determining thespectrum of such dyons. Our analysis leads to the correct contour prescriptionfor integrating the partition function to determine the spectrum in differentdomains of the moduli space separated by the walls of marginal stability.

Genus Two Surface and Quarter BPS Dyons: The Contour Prescription