This is just a short post to draw attention to a new preprint by Friedl and Powell The presentation of the Blanchfield pairing of a knot via a Seifert matrix.
The Blanchfield pairing on the Alexander module occurs in various places in knot theory, including in quantum topology. Levine’s 1977 argument for its expression in terms of the Seifert matrix doesn’t make easy reading (the authors suggest it’s incomplete- I can’t judge), and it is notoriously difficult to prove that the Blanchfield pairing is Hermitian. The authors deal deftly with both problems using a more modern but clearly sensible toolbox. Time to rewrite the textbooks.
I wish there were more papers like this. Some aspects of low dimensional topology could use a careful, sensible, modern reboot such as that of this paper.