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
The theory of discriminants shows that there are no non-trivial global etale extensions of
Okay, but then primes should correspond to certain submanifolds of
Hence, primes might be viewed as circles embedded in
because the algebraic fundamental groups of
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
Perhaps I should make a pod- or vod-cast of that 20 minute talk, one day…