Twisted Blanchfield pairings and twisted signatures I: Algebraic background

TL;DR

This paper establishes the algebraic framework for twisted linking forms and signatures of knots and three-manifolds, focusing on their classification and matrix representations, laying groundwork for subsequent research.

Contribution

It introduces the algebraic foundations for twisted linking forms, classifies them up to isometry and Witt equivalence, and explores their matrix representations.

Findings

Classified twisted linking forms up to isometry and Witt equivalence.

Developed methods to define and compute signature invariants.

Provided criteria for representing linking forms by matrices.

Abstract

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 $M\times M\to\mathbb{F}(t)/\mathbb{F}[t^{\pm1}]$ 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.