On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres

TL;DR

This paper classifies Alexander modules with Blanchfield forms for null-homologous knots in rational homology spheres, identifying when unique or multiple isomorphism classes occur, and constructs explicit examples for all cases.

Contribution

It provides a complete classification of Alexander modules and Blanchfield forms for these knots, including realization results for all classified modules.

Findings

Characterization of Alexander modules with Blanchfield forms

Identification of conditions for unique or multiple isomorphism classes

Explicit construction of knots realizing all classified modules

Abstract

In this article, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules A equipped with their Blanchfield forms $\phi$, and the modules A such that there is a unique isomorphism class of $(A,\phi)$, and we prove that for the other modules A, there are infinitely many such classes. We realise all these $(A,\phi)$ by explicit knots in rational homology spheres.