Twisted cohomology pairings of knots II; to classical invariants

TL;DR

This paper demonstrates how bilinear cup products with local coefficients in closed 3-manifolds can recover twisted pairings of infinite covers and Casson-Gordon signatures, providing diagrammatic computation methods.

Contribution

It introduces a novel approach linking cup products with local coefficients to classical invariants like twisted pairings and signatures, with diagrammatic computation techniques.

Findings

Recovered twisted pairings from cup products with local coefficients

Connected cup products to Casson-Gordon local signatures

Provided diagrammatic methods for computing pairings and signatures

Abstract

We show that bilinear cup products with local coefficients of closed 3-manifolds recover some twisted pairings of infinite covers and the Casson-Gordon local signatures. As a result, we further give diagrammatic computations of the pairings and signatures.