The existence of homologically fibered links and solutions of some equations

TL;DR

This paper explores the existence of homologically fibered links in 3-manifolds, relating it to algebraic invariants and equations, and computes a specific invariant for certain manifolds.

Contribution

It provides a new interpretation of homologically fibered links via algebraic equations and links this to the first homology group and torsion linking form.

Findings

Existence of homologically fibered links characterized by algebraic equations.

Computed the invariant hc for specific 3-manifolds with generator torsion linking forms.

Connected the existence problem to surgery diagrams and algebraic invariants.

Abstract

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.