# Twisted Blanchfield pairings, twisted signatures and Casson-Gordon invariants

### Abstract

This paper decomposes into two main parts. In the algebraic part, we prove an isometry classification of linking forms over $\mathbb{R}[t^{\pm1}]$ and $\mathbb{C}[t^{\pm1}]$. Using this result, we associate signature functions to any such linking form and thoroughly investigate their properties. The topological part of the paper applies this machinery to twisted Blanchfield pairings of knots. We obtain twisted generalizations of the Levine-Tristram signature function which share several of its properties. We study the behavior of these twisted signatures under satellite operations. In the case of metabelian representations, we relate our invariants to the Casson-Gordon invariants and obtain a concrete formula for the metabelian Blanchfield pairings of satellites. Finally, we perform explicit computations on certain linear combinations of algebraic knots, recovering a non-slice result of Hedden, Kirk and Livingston.

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