Twisted Blanchfield pairings and twisted signatures I: Algebraic background

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 $M\times M\to \mathbb{F}(t)/\mathbb{F}[t^{\pm 1}]$ 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.

35 pages, no figure. Contains Sections 2-5 of arXiv:1809.08791v1, which is now split into three papers: this is the first, the second is "Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants" (arXiv:1809.08791) and the third is "Twisted Blanchfield pairings and twisted signatures III: Applications". v2: to appear in Linear. Algebra. Appl

References in corpus (1)

• Double L-theory