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

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.