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In this article we determine the structure of a twisted first cohomologygroup of the first homology of a trivalent graph with a coefficient associatedwith the quantum Clebsch-Gordan condition. As an application we give acharacterization of a combinatorial property, the external edge condition,which is defined by the author in the study of the Heisenberg representation onthe TQFT-module.

External edge condition and group cohomologies associated with the quantum Clebsch-Gordan condition