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What is the knot associated to a prime?

Published January 2, 2011 by lieven

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…

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NeB : 7 years and now an iPad App

Published December 31, 2010 by lieven

Exactly 7 years ago I wrote my first post. This blog wasn’t called NeB yet and it used pMachine, a then free blogging tool (later transformed into expression engine), rather than WordPress.

Over the years NeB survived three hardware-upgrades of ‘the Matrix’ (the webserver hosting it), more themes than I care to remember, and a couple of dramatic closure announcements…

But then we’re still here, soldiering on, still uncertain whether there’s a point to it, but grateful for tiny tokens of appreciation.

Such as this morning’s story: Chandan deemed it necessary to correct two spelling mistakes in a 2 year old Fun-math post on Weil and the Riemann hypothesis (also reposted on Neb here). Often there’s a story behind such sudden comments, and a quick check of MathOverflow revealed this answer and the comments following it.

I thank Ed Dean for linking to the Fun-post, Chandan for correcting the misspellings and Georges for the kind words. I agree with Georges that a cut&copy of a blogpost-quoted text does not require a link to that post (though it is always much appreciated). It is rewarding to see such old posts getting a second chance…

Above the Google Analytics graph of the visitors coming here via a mobile device (at most 5 on a good day…). Anticipating much more iPads around after tonights presents-session I’ve made NeB more accessible for iPods, iPhones, iPads and other mobile devices.

The first time you get here via your Mac-device of choice you’ll be given the option of saving NeB as an App. It has its own icon (lowest row middle, also the favicon of NeB) and flashy start-up screen.

Of course, the whole point trying to make NeB more readable for Mobile users you get an overview of the latest posts together with links to categories and tags and the number of comments. Sliding through you can read the post, optimized for the device.

I do hope you will use the two buttons at the end of each post, the first to share or save it and the second to leave a comment.

I wish you all a lot of mathematical (and other) fun in 2011 :: lieven.

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Seating the first few billion Knights

Published December 30, 2010 by lieven

The odd Knight of the round table problem asks for a consistent placement of the n-th Knight in the row at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements.

The first identifies the multiplicative group of its non-zero elements with the group of all odd complex roots of unity, under complex multiplication. The second uses Conway’s ‘simplicity rules’ in ONAG to define an addition and multiplication on the set of all ordinal numbers.

Here’s the seating arrangement for the first 15 knights. Conway proved that all finite ordinals smaller than $2^{2^i} $ form a subfield of $\overline{\mathbb{F}}_2 $. The first non-trivial one being $\{ 0,1,2,3 \} $ with smallest multiplicative generator $2 $, whence we place Knight 2 at $e^{2 i \pi/3} $ and as $2^2=3 $ we know where to place the third Knight.

The next subfield is made of the numbers $\{ 0,1,2,\ldots,13,14,15 \} $ and its non-zero numbers form a cyclic group of order 15. Hence we need to find the smallest generator of this group satisfying the additional property of being compatible with the earlier seating, that is, its fifth power must equal to 2. Checking the multiplication table reproduced here one verifies that the wanted generator is 4 and so we place Knight 4 at $e^{\frac{2 \pi i}{15}} $ and as all the ordinals smaller than 16 are powers of 4 this tells us where to place the Knights until we get to the 15th in the row.

In february we were able to seat the first few thousand Knights by showing by hand that 32 is the smallest ordinal such that its 15-th power is equal to 4 and using SAGE that 1051 is the smallest ordinal whose 257-th power equals 32. These calculations enabled us to seat the Knights until we get to the 65536-th in the row.

Today I managed to show that 1361923 is the smallest ordinal such that its 65537-th power equals 1051. You can verify this statement in SAGE using the method explained in the february post


sage: R.< x,y,z,t,u >=GF(2)[]

sage: S.< a,b,c,d,e > =
R.quotient((x^2+x+1,y^2+y+x,z^2+z+x*y,t^2+t+x*y*z,u^2+u+x*y*z*t))

sage: (c*e+b*e+a*b*c*d+b*c*d+a*b*d+a+1)^65537
c^2 + b*d + a + 1

(It takes a bit longer to check minimality of 1361923). That is, we have to seat Knight 1361923 at $e^{\frac{2 \pi i}{4294967295}} $ and because all the numbers smaller than 4294967296 are powers of 1361923 we have seating arrangements for the first 4294967295 Knights!

I did try the same method in february but ran into time- and memory-problems on my 2.4Ghz 2Gb MacBook. Today I upgraded from Sage 3.3 to Sage 4.6 and this version is a lot faster (using the 64-bit architecture) and also appears to be much better at memory-management. Thank you, Sage-community!

Wishing you all a lot of mathematical (and other) fun in the prime-number year 2011.

atb :: lieven.

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