Factorization formulas and computations of higher-order Alexander invariants for homologically fibered knots

TL;DR

This paper investigates higher-order Alexander invariants of homologically fibered knots, providing factorization formulas and explicit calculations for specific non-fibered examples using metabelian quotients.

Contribution

It offers detailed analysis and concrete computations of higher-order Alexander invariants for homologically fibered knots, especially focusing on metabelian quotients.

Findings

Factorization of invariants into Magnus matrix and Reidemeister torsion parts

Explicit calculations for all 12-crossing non-fibered homologically fibered knots

Demonstration of invariants' behavior in specific knot classes

Abstract

Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey and Friedl are generally factorized into the part of the Magnus matrix and that of a certain Reidemeister torsion, both of which are known as invariants of homology cylinders over a surface. In this paper, we study more details of the invariants and give some concrete calculations by restricting to the case of the invariants associated with metabelian quotients of their knot groups. We provide examples of explicit calculations of the invariants for all the 12 crossings non-fibered homologically fibered knots.