Twisted Blanchfield Pairings and Twisted Signatures I: Algebraic Background

This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate signature invariants associated to a linking form $M \times M \to \mathbb{F}(t) / \mathbb{F}[t^{\pm 1}]$ for $\mathbb{F} = \mathbb{R}, \mathbb{C}$ , where $M$ is a torsion $\mathbb{F}[t^{\pm 1}]$ -module. Along the way, we classify such linking forms up to isometry and Witt equivalence and study whether they can be represented by matrices.