  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.
