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# An Explicit Computation of the Blanchfield Pairing for Arbitrary Links

Published:2018-02-26
Printed: Oct 2018
• Anthony Conway,
Université de Genève, Section de mathématiques, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland
 Format: LaTeX MathJax PDF

## 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 $\operatorname{Bl}(L)$.
 Keywords: link, Blanchfield pairing, C-complex, Alexander module
 MSC Classifications: 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45}

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