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

EXTERNAL EDGE CONDITION AND GROUP COHOMOLOGIES ASSOCIATED WITH THE QUANTUM CLEBSCH-GORDAN CONDITION