the birthday of the primes=knots analogy

Last time we discovered that the mental picture to view prime numbers as knots in $S^3$ was first dreamed up by David Mumford. Today, we’ll focus on where and when this happened.

3. When did Mazur write his unpublished preprint?

According to his own website, Barry Mazur did write the paper Remarks on the Alexander polynomial in 1963 or 1964. A quick look at the references gives us a coarse lower- and upper-estimate.

Apart from a paper by Iwasawa and one by Milnor, all references predate 1962 giving us a lower-bound. More interesting is reference (14) to David Mumford’s Geometric Invariant Theory (GIT) which was first published in 1965 and is referred to as ‘in preparation’, so the paper was written no later than 1965. If we look a bit closer we see than some GIT-references are very precise

indicating that Mazur must have had the final version of GIT to consult, making it rather difficult to believe that the preprint was written late 1963 or early 1964.

Mazur’s dating of the preprint is probably based on this penciled note on the frontpage of the only surviving copy of the preprint

It reads : “Date from about 63/64, H.R. Morton”. Hugh Morton of Liverpool University confirms that it is indeed his writing on the preprint.

Further, he told me that early 64 Christopher Zeeman held a Topology Symposium in Cambridge UK, where Hugh was a graduate student at the time and, as far as he could recall, Mazur attended that conference and gave him the preprint on that occasion, whence the 63/64 dating. Hugh kindly offered to double-check this with Terry Wall who cannot remember Mazur attending that particular conference.

In fact, we will see that a more correct dating of the Mazur-preprint will be : late 1964 or early 1965.

4. The birthday : July 10th 1964

Clearly, Mumford’s insight predates the Mazur-preprint. In the first section, Mazur mentions ‘Grothendieck cohomology groups’ rather than ‘Etale cohomology groups’.

At the time, Artin’s seminar notes on Grothendieck topologies (spring 1962) were widely distributed, and Artin and Grothendieck were in the process of developing etale cohomology in their Paris 1963/64 seminar SGA 4, while Mumford was working on GIT in Harvard.

Mike Artin, David Mumford and Jean-Louis Verdier all attended the Woods Hole conference from july 6 till july 31 1964, famous for producing the Atiyah-Bott fixed point theorem (according to Fulton first proved by Verdier at the conference).

Etale cohomology was a hot topic at that conference. On july 10th there were three talks, Artin spoke on ‘Etale cohomology of schemes’, Verdier on ‘A duality theorem in the etale cohomology of schemes’ and John Tate on ‘Etale cohomology over number fields’.

After a first week of talks, more informal seminars were organized, including the Atiyah-Bott seminar leading to the ‘Woods hole duality theorem’ and one by Lubin-Tate and Serre on elliptic curves and formal groups. Two seminars adressed Etale Cohomology.

Artin and Verdier ran a seminar on the etale cohomology of number fields leading to their duality result, and, three young turks : Daniel Quillen, Steve Kleiman and Robin Hartshorne ran a Baby Seminar on Etale cohomology

Probably it is safe to say that the talks by Artin, Verdier and Tate on July 10th sparked the primes=knots idea, and if not then, a couple of days later.

5. The birthplace : the Whitney Estate

The ‘Woods Hole’ conference took place at the Whitney Estate and all the lectures took place in the rustic rooms of the main building and the participants (and their families) were housed in rented cottages in the neighborhood, for the duration of the summer.

The only picture i managed to find from the Whitney house comes from a rather surprising source : Gardeners and Caretakers ofWoods Hole. Anyway, here it is :

Probably, the knots=primes analogy was first dreamed up inside, or in the immediate neighborhood, on a walk to or from the cottages, overlooking the harbor.

Who dreamed up the primes=knots analogy?

One of the more surprising analogies around is that prime numbers can be viewed as knots in the 3-sphere $S^3$. The motivation behind it is that the (etale) fundamental group of $\pmb{spec}(\mathbb{Z}/(p))$ is equal to (the completion) of the fundamental group of a circle $S^1$ and that the embedding

$\pmb{spec}(\mathbb{Z}/(p)) \subset \pmb{spec}(\mathbb{Z})$

embeds this circle as a knot in a 3-dimensional simply connected manifold which, after Perelman, has to be $S^3$. For more see the what is the knot associated to a prime?-post.

In recent months new evidence has come to light allowing us to settle the genesis of this marvelous idea.

1. The former consensus

Until now, the generally accepted view (see for example the ‘Mazur-dictionary-post’ or Morishita’s expository paper) was that the analogy between knots and primes was first pointed out by Barry Mazur in the middle of the 1960’s when preparing for his lectures at the Summer Conference on Algebraic Geometry, at Bowdoin, in 1966. The lecture notes where later published in 1973 in the Annales of the ENS as ‘Notes on etale cohomology of number fields’.

For further use in this series of posts, please note the acknowledgement at the bottom of the first page, reproduced below : “It gives me pleasure to thank J.-P. Serre for his vigorous editing and his suggestions and corrections, which led to this revised version.”

Independently, Yuri I. Manin spotted the same analogy at around the same time. However, this point of view was quickly forgotten in favor of the more classical one of viewing number fields as analogous to algebraic function fields of one variable. Subsequently, in the mid 1990’s Mikhail Kapranov and Alexander Reznikov took up the analogy between number fields and 3-manifolds again, and called the resulting study arithmetic topology.

2. The new evidence

On december 13th 2010, David Feldman posted a MathOverflow-question Mazur’s unpublished manuscript on primes and knots?. He wrote : “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.

Two months later, on february 15th David Feldman suddenly found the missing preprint in his mail-box and made it available. The preprint is now also available from Barry Mazur’s website. Mazur adds the following comment :

“In 1963 or 1964 I wrote an article Remarks on the Alexander Polynomial [PDF] about the analogy between knots in the three-dimensional sphere and prime numbers (and, correspondingly, the relationship between the Alexander polynomial and Iwasawa Theory). I distributed some copies of my article but never published it, and I misplaced my own copy. In subsequent years I have had many requests for my article and would often try to search through my files to find it, but never did. A few weeks ago Minh-Tri Do asked me for my article, and when I said I had none, he very kindly went on the web and magically found a scanned copy of it. I’m extremely grateful to Minh-Tri Do for his efforts (and many thanks, too, to David Feldman who provided the lead).”

The opening paragraph of this unpublished preprint contains a major surprise!

Mazur points to David Mumford as the originator of the ‘primes-are-knots’ idea : “Mumford has suggested a most elegant model as a geometric interpretation of the above situation : $\pmb{spec}(\mathbb{Z}/p\mathbb{Z})$ is like a one-dimensional knot in $\pmb{spec}(\mathbb{Z})$ which is like a simply connected three-manifold.”

In a later post we will show that one can even pinpoint the time and place when and where this analogy was first dreamed-up to within a few days and a couple of miles.

For the impatient among you, have a sneak preview of the cradle of birth of the primes=knots idea…

changes (ahead)

In view or recents events & comments, some changes have been made or will be made shortly :

categories : Sanitized the plethora of wordpress-categories to which posts belong. At the moment there are just 5 categories : ‘stories’ and ‘web’ (for all posts with low math-content) and three categories ‘level1’, ‘level2’ and ‘level3’, loosely indicating the math-difficulty of a post.

MathJax : After years of using LatexRender and WP-Latex, we’ll change to MathJax from now on. I’ll try to convert older posts as soon as possible. (Update : did a global search and replace. ‘Most’ LaTeX works, major exceptions being matrices and xymatrix commands. I’ll try to fix those later with LatexRender.)

theme : The next couple of days, the layout of this site may change randomly as I’ll be trying out things with the Swift wordpress theme. Hopefully, this will converge to a new design by next week.

name : Neverendingbooks will be renamed to something more math-related. Clearly, the new name will depend on the topics to be covered. On the main index page a pop-up poll will appear in the lower right-hand corner after 10 seconds. Please fill in the topics you’d like us to cover (no name or email required).

This poll will close on friday 21st at 12 CET and its outcome will influence name/direction of this blog. Use it also if you have a killer newname-suggestion. Among the responses so far, a funnier one : “An intro to, or motivation for non-commutative geometry, aimed at undergraduates. As a rule, I’d take what you think would be just right for undergrads, and then trim it down a little more.”

guest-posts : If you’d like to be a guest-blogger here at irregular times, please contact me. The first guest-post will be on noncommutative topology and the interpretation of quantum physics, and will appear soon. So, stay tuned…

the Reddit (after)effect

Sunday january 2nd around 18hr NeB-stats went crazy.

Referrals clarified that the post ‘What is the knot associated to a prime?’ was picked up at Reddit/math and remained nr.1 for about a day.

Now, the dust has settled, so let’s learn from the experience.

A Reddit-mention is to a blog what doping is to a sporter.

You get an immediate boost in the most competitive of all blog-stats, the number of unique vistors (blue graph), but is doesn’t result in a long-term effect, and, it may even be harmful to more essential blog-stats, such as the average time visitors spend on your site (yellow graph).

For NeB the unique vistors/day fluctuate normally around 300, but peaked to 1295 and 1733 on the ‘Reddit-days’. In contrast, the avg. time on site is normally around 3 minutes, but dropped the same days to 44 and 30 seconds!

Whereas some of the Reddits spend enough time to read the post and comment on it, the vast majority zap from one link to the next. Having monitored the Reddit/math page for two weeks, I’m convinced that post only made it because it was visually pretty good. The average Reddit/math-er is a viewer more than a reader…

So, should I go for shorter, snappier, more visual posts?

Let’s compare Reddits to those coming from the three sites giving NeB most referrals : Google search, MathOverflow and Wikipedia.

This is the traffic coming from Reddit/math, as always the blue graph are the unique visitors, the yellow graph their average time on site, blue-scales to the left, yellow-scales to the right.

Here’s the same graph for Google search. The unique visitors/day fluctuate around 50 and their average time on site about 2 minutes.

The math-related search terms most used were this month : ‘functor of point approach’, ‘profinite integers’ and ‘bost-connes sytem’.

More rewarding to me are referrals from MathOverflow.

The number of visitors depends on whether the MathO-questions made it to the front-page (for example, the 80 visits on december 15, came from the What are dessins d’enfants?-topic getting an extra comment that very day, and having two references to NeB-posts : The best rejected proposal ever and Klein’s dessins d’enfant and the buckyball), but even older MathO-topics give a few referrals a day, and these people sure take their time reading the posts (+ 5 minutes).

Other MathO-topics giving referrals this month were Most intricate and most beautiful structures in mathematics (linking to Looking for F-un), What should be learned in a first serious schemes course? (linking to Mumford’s treasure map (btw. one of the most visited NeB-posts ever)), How much of scheme theory can you visualize? (linking again to Mumford’s treasure map) and Approaches to Riemann hypothesis using methods outside number theory (linking to the Bost-Connes series).

Finally, there’s Wikipedia

giving 5 to 10 referrals a day, with a pretty good time-on-site average (around 4 minutes, peaking to 12 minutes). It is rewarding to see NeB-posts referred to in as diverse Wikipedia-topics as ‘Fifteen puzzle’, ‘Field with one element’, ‘Evariste Galois’, ‘ADE classification’, ‘Monster group’, ‘Arithmetic topology’, ‘Dessin d’enfant’, ‘Groupoid’, ‘Belyi’s theorem’, ‘Modular group’, ‘Cubic surface’, ‘Esquisse d’un programme’, ‘N-puzzle’, ‘Shabat polynomial’ and ‘Mathieu group’.

What lesson should be learned from all this data? Should I go for shorter, snappier and more visual posts, or should I focus on the small group of visitors taking their time reading through a longer post, and don’t care about the appallingly high bounce rate the others cause?

What is the knot associated to a prime?

Sometimes a MathOverflow question gets deleted before I can post a reply…

Yesterday (New-Year) PD1&2 were visiting, so I merely bookmarked the What is the knot associated to a prime?-topic, promising myself to reply to it this morning, only to find out that the page no longer exists.

From what I recall, the OP interpreted one of my slides of the April 1st-Alumni talk

as indicating that there might be a procedure to assign to a prime number a specific knot. Here’s the little I know about this :

Artin-Verdier duality in etale cohomology suggests that $Spec(\mathbb{Z}) $ is a 3-dimensional manifold, as Barry Mazur pointed out in this paper

The theory of discriminants shows that there are no non-trivial global etale extensions of $Spec(\mathbb{Z}) $, whence its (algebraic) fundamental group should be trivial. By Poincare-Perelman this then implies that one should view $Spec(\mathbb{Z}) $ as the three-sphere $S^3 $. Note that there is no ambiguity in this direction. However, as there are other rings of integers in number fields having trivial fundamental group, the correspondence is not perfect.

Okay, but then primes should correspond to certain submanifolds of $S^3 $ and as the algebraic fundamental group of $Spec(\mathbb{F}_p) $ is the profinite completion of $\mathbb{Z} $, the first option that comes to mind are circles

Hence, primes might be viewed as circles embedded in $S^3 $, that is, as knots! But which knots? Well, as far as I know, nobody has a procedure to assign a knot to a prime number, let alone one having p crossings. What is known, however, is that different primes must correspond to different knots

because the algebraic fundamental groups of $Spec(\mathbb{Z})- { p } $ differ for distinct primes. This was the statement I wanted to illustrate in the first slide.

But, the story goes a lot further. Knots may be linked and one can detect this by calculating the link-number, which is symmetric in the two knots. In number theory, the Legendre symbol, plays a similar role thanks to quadratic reciprocity

and hence we can view the Legendre symbol as indicating whether the knots corresponding to different primes are linked or not. Whereas it is natural in knot theory to investigate whether collections of 3, 4 or 27 knots are intricately linked (or not), few people would consider the problem whether one collection of 27 primes differs from another set of 27 primes worthy of investigation.

There’s one noteworthy exception, the Redei symbol which we can now view as giving information about the link-behavior of the knots associated to three different primes. For example, one can hunt for prime-triples whose knots link as the Borromean rings

(note that the knots corresponding to the three primes are not the unknot but more complicated). Here’s where the story gets interesting : in number-theory one would like to discover ‘higher reciprocity laws’ (for collections of n prime numbers) by imitating higher-link invariants in knot-theory. This should be done by trying to correspond filtrations on the fundamental group of the knot-complement to that of the algebraic fundamental group of $Spec(\mathbb{Z})-{ p } $ This project is called arithmetic topology

Perhaps I should make a pod- or vod-cast of that 20 minute talk, one day…

Lists 2010 : MathOverflow bookmarks

A few MathOverflow threads I bookmarked in 2010 for various reasons.