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There is one generalization of fibered links in 3-manifolds, calledhomologically fibered links. It is known that the existence of homologicallyfibered links whose fiber surface has a given homeomorphic type is determinedby the first homology group and its torsion linking form of the ambient3-manifold. In this paper, we interpret the existence of homologically fiberedlinks with that of a solution of some equation, in terms of the first homologygroup and its torsion linking form or a surgery diagram of the ambientmanifold. As an application, we compute the ${\rm hc}(\cdot)$, defined throughhomologically fibered knots, for 3-manifolds whose torsion linkng formsrepresent a generator of linkings.

The existence of homologically fibered links and solutions of some equations