The Atiyah-Patodi-Singer rho invariant and signatures of links

TL;DR

This paper develops a cut-and-paste formula for the Atiyah-Patodi-Singer rho invariant to relate it to link signatures, providing new tools to analyze 3-manifolds and link invariants.

Contribution

It introduces a versatile formula for the rho invariant, enabling intrinsic descriptions of link signatures and analysis of manifolds obtained via Dehn surgery.

Findings

The rho invariant can be expressed as a sum involving the link signature.

A new method to compute rho invariants for manifolds from Dehn surgeries.

Connections established between rho invariants and classical link signatures.

Abstract

Relations between the Atiyah-Patodi-Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link $L$ as the rho invariant of some closed $3$-manifold $Y_L$ intrinsically associated to $L$. We study then the rho invariant of the manifolds obtained by Dehn surgery on a $L$ along integer and rational framings. Inspired by related results of Casson and Gordon and Cimasoni and Florens, we give formulas expressing this value as a sum of the multivariable signature of $L$ and some easy-to-compute extra terms.