Peripheral elements in reduced Alexander modules

TL;DR

This paper explores the structure of reduced Alexander modules for classical and virtual links, showing that enhancing these modules with peripheral elements yields a stronger invariant capable of determining all linking numbers.

Contribution

It introduces a method to enhance reduced Alexander modules by incorporating peripheral elements, significantly improving their ability to distinguish links.

Findings

Enhanced modules determine all linking numbers in a link

Peripheral elements provide a stronger link invariant

Modules alone cannot detect zero linking numbers

Abstract

We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander module by singling out these peripheral elements provides a significantly stronger link invariant. In particular, the enhanced module determines all linking numbers in a link; in contrast, the module alone does not even detect how many linking numbers are $0$.