The first Alexander Z[Z]-modules of surface-links and of virtual links

TL;DR

This paper characterizes the first Alexander Z[Z]-modules of surface-links and virtual links, providing classifications based on components and genus, and introduces a graded structure across various link types.

Contribution

It offers a comprehensive characterization of the first Alexander Z[Z]-modules for surface-links and virtual links, including estimates and graded structures, extending previous results.

Findings

Characterization of modules fixing components and genus

Estimation of total genus from modules

Graded structure on Alexander modules across link types

Abstract

We characterize the first Alexander Z[Z]-modules of ribbon surface-links in the 4-sphere fixing the number of components and the total genus, and then the first Alexander Z[Z]-modules of surface-links in the 4-sphere fixing the number of components. Using the result of ribbon torus-links, we also characterize the first Alexander Z[Z]-modules of virtual links fixing the number of components. For a general surface-link, an estimate of the total genus is given in terms of the first Alexander Z[Z]-module. We show a graded structure on the first Alexander Z[Z]-modules of all surface-links and then a graded structure on the first Alexander Z[Z]-modules of classical links, surface-links and higher-dimensional manifold-links.