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

TL;DR

This paper extends the theory of knot invariants by defining twisted signature functions associated with representations of the knot group, linking them to classical signatures and Casson-Gordon invariants, and providing satellite formulas.

Contribution

It introduces a new class of twisted signature invariants for knots, generalizing classical signatures and relating them to Casson-Gordon invariants for specific representations.

Findings

Twisted signature function generalizes Levine-Tristram signature.

For abelian representations, it recovers classical signatures.

For metabelian representations, it relates 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.