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 ${\sigma }_{K,\rho }:S^1\to \mathbb{Z}$. 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.

v1: 83 pages. v2: expanded introduction. v3 contains Sections 6,7 and 8 of the original paper. Sections 2-5 are now in "Twisted Blanchfield pairings and twisted signatures I: Algebraic background" while Section 9 is now in "Twisted Blanchfield pairings and twisted signatures III: Applications". In v4 an appendix has been added: it contains a detailed proof of Corollary 4.6. 43 pages, no figures