  We discuss the fundamental (relative) 3-classes of knots (or hyperbolic links), and provide diagrammatic descriptions of the push-forwards with respect to every link-group representation. The point is an observation of a bridge between the relative group homology and quandle homology from the viewpoints of Inoue--Kabaya map \cite{IK}. Furthermore, we give an algorithm to algebraically describe the fundamental 3-class of any hyperbolic knot. 
On the fundamental 3-classes of knot group representations