Paper detail

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

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 $ρ$ of the knot group, we define a twisted signature function $σ_{K,ρ} \colon S^1 \to \mathbb{Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $σ_K$. When the representation is abelian, $σ_{K,ρ}$ recovers $σ_K$, while for appropriate metabelian representations, $σ_{K,ρ}$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $σ_{K,ρ}$ and for twisted Blanchfield forms.