  There is one generalization of fibered links in 3-manifolds, called
homologically fibered links. It is known that the existence of homologically
fibered links whose fiber surface has a given homeomorphic type is determined
by the first homology group and its torsion linking form of the ambient
3-manifold. In this paper, we interpret the existence of homologically fibered
links with that of a solution of some equation, in terms of the first homology
group and its torsion linking form or a surgery diagram of the ambient
manifold. As an application, we compute the ${\rm hc}(\cdot)$, defined through
homologically fibered knots, for 3-manifolds whose torsion linkng forms
represent a generator of linkings.
