Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants

Abstract

This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot $K$ and a representation $\rho$ of the knot group, we define a twisted signature function $\sigma_{K,\rho} \colon S^1 \to \mathbb{Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $\sigma_K$. When the representation is abelian, $\sigma_{K,\rho}$ recovers $\sigma_K$, while for appropriate metabelian representations, $\sigma_{K,\rho}$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $\sigma_{K,\rho}$ and for twisted Blanchfield forms.