An explicit computation of the Blanchfield pairing for arbitrary links

Abstract

Given a link $L$, the Blanchfield pairing $\mathrm{Bl}(L)$ 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, $\mathrm{Bl}(L)$ 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 $\mathrm{Bl}(L)$.

23 pages, 1 figure. v2: Corollary 1.2 has been weakened; an appendix containing the proof of Lemma 3.1 has been added; other minor changes. To appear in the Canadian Journal of Mathematics. v3: The definition of the pairing ${\lambda }_H$ has been corrected

References in corpus (1)

• Topological minimal genus and $L^2$-signatures

Cited by in corpus (4)

• Twisted Blanchfield pairings and twisted signatures I: Algebraic background
• Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants
• Untwisting number and Blanchfield pairings
• On Arf invariants of colored links