# An explicit computation of the Blanchfield pairing for arbitrary links

## TL;DR

This paper provides a general explicit formula for computing the Blanchfield pairing for any link, extending previous special cases and confirming its hermitian property.

## Contribution

It introduces a universal method to compute the Blanchfield pairing for arbitrary links, generalizing prior specific cases and simplifying proofs of its properties.

## Key findings

- Explicit formula for Blanchfield pairing of any link
- Proof that the pairing is hermitian
- Simplified proofs of key properties of the pairing

## Abstract

Given a link $L$, the Blanchfield pairing $\operatorname{Bl}(L)$ is a pairing which is defined on the torsion submodule of the Alexander module of $L$. In some particular cases, namely if $L$ is a boundary link or if the Alexander module of $L$ is torsion, $\operatorname{Bl}(L)$ can be computed explicitly; however no formula is known in general. In this article, we compute the Blanchfield pairing of any link, generalizing the aforementioned results. As a corollary, we obtain a new proof that the Blanchfield pairing is hermitian. Finally, we also obtain short proofs of several properties of $\operatorname{Bl}(L)$.