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We comprehensively show the topological structure--order of zeros felt by anelectron at the positions of other electrons --for the enigmatic filling factor5/2 at the "Pfaffian-shift" in a spherical geometry. A set of linearlyindependent antisymmetric functions that are constructed from the possiblegraphs preserving this topological structure provides complete basis fordetermining an accurate ground state wave function for any interaction, inparticular the Coulomb interaction. One of the graphs describes $Z_2$para-fermion equivalently the Pfaffian wave function, but some of the othergraphs also contribute to form the exact ground state for the Coulombinteraction. We further show that our wave function for the Coulomb interactionsupports clustering up to half of the composite bosons, signifying astrong-coupling regime.

Topological Structure and an Accurate Wave Function for the Enigmatic 5/2 Fractional Quantum Hall State