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The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur.

I'm not absolutely sure this is the one I mean, but in his paper, Analogies between group actions on 3-manifolds and number fields, Adam Sikora cites

B. Mazur, Remarks on the Alexander polynomial, unpublished notes.

He also cites the published paper

B. Mazur, Notes on étale topology of number fields, Ann. Sci. Ecole Norm. Sup. (4)6 (1973), 521-552.

I suppose an expert would recognize as relevance of this paper, but I don't see that even the word "knot" ever occurs there.

[My Question] Does anyone have a copy of Mazur's note that they would share, please? If not, has anyone at least actually seen it.

By the way, already years ago, I asked Mazur himself. I watched him kindly search his office, but he came up dry.

I realize that whatever insights the original note contains have doubtless been surpassed after c. 40 years by published results, but historical curiosity drives my desire to see the document that started the industry.

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    $\begingroup$ When I asked him, he gave me a copy of this fascinating article: books.google.com/… $\endgroup$
    – Thomas Riepe
    Commented Dec 13, 2010 at 22:08
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    $\begingroup$ The most au courant version of this analogy, which I learned from Deninger (though I don't know whether it originates with him) is that a number field is like a 3-manifold with a flow, in which the primes are the closed orbits! $\endgroup$
    – JSE
    Commented Mar 6, 2011 at 19:44
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    $\begingroup$ @JSE: some written references would be highly appreciated. $$ $$ $\endgroup$
    – Chandan Singh Dalawat
    Commented Mar 7, 2011 at 3:41

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This showed up in my snail-mail today, so I'm sharing the wealth:

http://ifile.it/rodc5is/mazur.pdf

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    $\begingroup$ Any interest in a stone-soup effort to Tex the beast? Ten volunteers, 3 pages each, say? $\endgroup$
    – David Feldman
    Commented Feb 17, 2011 at 4:14
  • $\begingroup$ That's a really good idea! Maybe open a meta for it? In any case, you can count me in. $\endgroup$
    – Dror Speiser
    Commented Mar 4, 2011 at 12:56
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    $\begingroup$ The link is dead. $\endgroup$
    – Anton Geraschenko
    Commented Oct 21, 2011 at 2:06
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    $\begingroup$ Hi Anton, I have re-upped: ifile.it/t5qp34u NB Since Mazur is distributing this preprint now from his own page, I don't plan trying to keep this fresh in perpetuity. $\endgroup$
    – David Feldman
    Commented Oct 22, 2011 at 6:27
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    $\begingroup$ @DavidFeldman Could you perhaps change the link to the location where Mazur himself is distributing the preprint? Your link has, of course, expired (and I wasn't sure which link on Mazur's site is the right one...). Could this be the right one? $\endgroup$
    – Danu
    Commented Oct 14, 2015 at 18:15
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$\newcommand\Z{\mathbf Z} \newcommand\F{\mathbf F} \newcommand\S{\mathbf S}$

Sadly, I don't have a copy of the unpublished note, but at least I can tell you what the published paper has to do with knots.

In that paper, Mazur shows that etale cohomology for Spec($\Z$) satisfies a sort of Poincare-duality as known for 3-dimensional manifolds. So, one might consider Spec($\Z$) as a 3-dimensional manifold. As its fundamental group is trivial, it should be after Poincare-Perelman the 3-sphere. Primes should be closed submanifolds in it and as the fundamental group of $\F_p=\Z/p\Z$ is $\hat\Z$, one might consider primes as circles embedded in $\S^3$, that is, as knots.

The analogy goes a lot deeper and I've written a blog-post about it some time ago (with references to Mazur's paper and follow-ups) :

Mazur's knotty dictionary

I've also given a talk about it, intended for a general public, last year. The slides are sadly in Dutch, but perhaps they convey what I was trying to tell : (34Mb download though ...)

What does a prime number look like?

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  • $\begingroup$ Dear Lieven, thanks for the answer, and great to see you back! $\endgroup$
    – Hailong Dao
    Commented Dec 13, 2010 at 21:48
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Thanks to this MO question, the paper Remarks on the Alexander Polynomial is now available on Barry Mazur's website.

Gratuitous addendum. For a recent overview, see

Morishita (Masanori), Analogies between prime numbers and knots, Sūgaku 58 (2006), no. 1, 40–63.

An English translation has appeared in

Sugaku Expositions 23 (2010), no. 1.

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