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

TL;DR

This paper investigates the twisted first cohomology group related to the quantum Clebsch-Gordan condition on trivalent graphs, providing insights into the external edge condition in TQFT modules.

Contribution

It determines the structure of a specific twisted cohomology group and characterizes the external edge condition in the context of quantum topology.

Findings

Structured the twisted first cohomology group of a graph's homology.

Characterized the external edge condition in TQFT modules.

Linked cohomological properties to combinatorial graph conditions.

Abstract

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.