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.
Categories: math.GT