Knot Theory – What is the Knot Associated to a Prime?

knot-theorynt.number-theory

I can't help but ask this question, having found out about arithmetic topology here on MO. There is a concise description of the MKR dictionary central to this philosophy here. This dictionary is used to translate statements in three-manifold topology into statements in number theory, and vice versa. Often, but not always, the resulting translated statements turn out to be true.

Before asking the following dumb question, let me be absolutely clear that I understand that the general analogy between knots and primes is complicated, and isn't perfect but only fruitful.

That said, I simply cannot help asking the following (certainly stupid) question.

I thank Theo Buehler for his comment below, as it helps to more directly ask the question I wanted to ask. I'll incorporate his comment here as part of the question:

According to p.24 in Lieven LeBruyn's slides (35 MB!), the trefoil knot is associated to 3. LeBruyn also gives pictures for 2,5 and 7. Is there, associated to the MKR analogy, a natural way to assign a knot to every prime?

This, I imagine, should not be possible in any reasonable way. I ask with the dim hope that I may be wrong.

Must the knot associated to 3 be the trefoil, as shown in LeBruyn's slides?

Best Answer

Jon,

I should have been more careful in drawing the slide mentioned above (and reproduced below). I didn't want to convey the idea that there is a method to assign to a knot a specified knot (and certainly not one having exactly p crossings...). I only wanted to depict the result that different primes should correspond to different knots.

(source)

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

(source)

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

(source)

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

(source)

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

(source)

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

(source)

(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

(source)

(my thanks to the KnotPlot site for the knot-pictures used in the slides)

(my apologies for cross-posting. MathOverflow was the intended place to reply to Jon's question, but this topic was temporarily removed yesterday.)

Related Solutions

[Math] Does this knot invariant distinguish trefoil chiralities

I'm very curious where this came up. In any case, the answer to the first question is yes, it does distinguish these trefoils; you found the minimal representatives.

Let $a_0,\dots,a_{N-1}$ be the roots of unity that are visited along the knot, in (cyclic) order. Suppose we have a minimal representative for some non-trivial knot. Then we cannot have $|a_k - a_{k+1}| = 1$ for any $k$, as otherwise we could replace this pair $a_k, a_{k+1}$ by a single root of unity (for $N-1$), adjusting the other roots of unity as appropriate. A little more subtly, we cannot have $|a_{k-1} - a_{k+1}| = 1$ either, as then we could again delete $a_k$ from the sequence to get a smaller representation. With these simple constraints, the smallest possible sequence for a non-trivial knot is the one you found for one of the trefoils with $N=7$. There are several possibilities for $N=8$, including the one you found for the other trefoil. I've included a very short Haskell program below that computes this. The possibilities for $N=8$ are $$ (2,7,5,3,1,6,4,0)\quad (2,5,7,3,1,6,4,0)\quad (3,6,1,4,7,2,5,0)\quad (2,6,4,1,7,3,5,0) $$ $$ (3,1,6,4,2,7,5,0)\quad (2,4,6,1,3,7,5,0)\quad (3,5,1,7,4,2,6,0)\quad (4,2,7,5,1,3,6,0) $$ $$ (3,1,5,7,2,4,6,0)\quad (5,3,1,6,4,2,7,0)\quad (2,4,6,1,3,5,7,0) $$

For the second question, I have never heard of this representation before.

Here is the code, for anyone interested.

 -- A (partial) circular stick representation is a list of integers,
 -- the order of the roots of unity to visit in order
 type CircStick = [Int]

 -- The next element ak after a partial representation a1, ..., a{k-1} 
 -- must satisfy
 --   (a) ak has not already been seen
 --   (b) |ak - a{k-1}| > 1
 --   (c) |ak - a{k-2}| > 1
 -- There are a few more "easy" constraint, eg the first and last entries
 -- cannot differ by one.  We do not impose those constraint here.
 nexts :: Int -> CircStick -> [Int]
 nexts n [] = [0]
 nexts n [a1] = filter (\a -> abs (a-a1) > 1) [0..n-1]
 nexts n (a1:a2:as) =
   filter (\a -> not (elem a as)) $
       filter (\a -> abs (a-a1) > 1) $
   filter (\a -> abs (a-a2) > 1) $
     [1..n-1]

 completions :: Int -> CircStick -> [CircStick]
 completions n as | length as >= n = [as]
 completions n as =
   concat [completions n (a:as) | a <- nexts n as]

 -- Impose final constraints:
 --    (a) Last entry cannot be 1
 --    (b) Take entry that is lexicographically less than its reverse
 --    (c) first and next-to-last entries cannot differ by one
 circSticks :: Int -> [CircStick]
 circSticks n = 
   filter (\as -> abs ((as!!0) - (as!!(n-2))) > 1) $
       filter (\as -> as < tail (reverse as)) $
   filter (\as -> head as /= 1) $
   (completions n [])

Edit: For those interested, here are the 108 possibilities for $N=9$. I hope there's some way of checking what these are more efficiently than just going through them by hand.

[[2,7,5,3,8,1,6,4,0],[2,7,5,3,1,8,6,4,0],[2,5,7,3,1,8,6,4,0],[2,7,5,1,3,8,6,4,0],[2,5,7,1,3,8,6,4,0],[2,6,8,3,5,1,7,4,0],[2,7,5,3,1,6,8,4,0],[2,5,7,3,1,6,8,4,0],[2,7,5,1,3,6,8,4,0],[2,5,7,1,3,6,8,4,0],[3,8,6,1,4,7,2,5,0],[3,6,8,1,4,7,2,5,0],[3,7,1,4,6,8,2,5,0],[2,8,6,4,1,7,3,5,0],[2,6,8,4,1,7,3,5,0],[2,7,4,1,6,8,3,5,0],[2,4,8,6,3,1,7,5,0],[2,4,6,8,3,1,7,5,0],[3,8,1,6,4,2,7,5,0],[3,1,8,6,4,2,7,5,0],[3,6,1,8,4,2,7,5,0],[3,1,6,8,4,2,7,5,0],[2,8,4,6,1,3,7,5,0],[2,4,8,6,1,3,7,5,0],[2,6,4,8,1,3,7,5,0],[2,4,6,8,1,3,7,5,0],[3,7,1,4,6,2,8,5,0],[2,7,4,1,6,3,8,5,0],[3,8,5,1,7,4,2,6,0],[3,7,5,1,8,4,2,6,0],[3,5,7,1,4,8,2,6,0],[4,7,1,3,5,8,2,6,0],[4,1,7,3,5,8,2,6,0],[4,8,2,5,7,1,3,6,0],[4,7,2,5,8,1,3,6,0],[4,2,7,5,1,8,3,6,0],[2,4,7,1,5,8,3,6,0],[2,8,5,3,7,1,4,6,0],[2,7,5,3,8,1,4,6,0],[3,8,1,5,7,2,4,6,0],[3,7,1,5,8,2,4,6,0],[2,7,5,3,1,8,4,6,0],[2,5,7,3,1,8,4,6,0],[3,1,7,5,2,8,4,6,0],[4,2,7,5,3,1,8,6,0],[3,5,1,7,4,2,8,6,0],[4,2,7,5,1,3,8,6,0],[2,4,7,1,5,3,8,6,0],[3,1,5,7,2,4,8,6,0],[5,8,3,1,6,4,2,7,0],[5,3,8,1,6,4,2,7,0],[3,5,8,1,6,4,2,7,0],[5,3,1,8,6,4,2,7,0],[3,5,1,8,6,4,2,7,0],[5,1,3,8,6,4,2,7,0],[5,3,1,6,8,4,2,7,0],[5,1,3,6,8,4,2,7,0],[4,6,1,3,8,5,2,7,0],[4,1,6,3,8,5,2,7,0],[4,6,2,8,5,1,3,7,0],[5,8,2,4,6,1,3,7,0],[5,2,8,4,6,1,3,7,0],[2,6,4,8,1,5,3,7,0],[4,2,6,8,1,5,3,7,0],[2,4,6,8,1,5,3,7,0],[2,6,4,1,8,5,3,7,0],[2,4,6,1,8,5,3,7,0],[2,5,8,3,6,1,4,7,0],[5,3,1,8,6,2,4,7,0],[3,5,1,8,6,2,4,7,0],[5,1,3,8,6,2,4,7,0],[5,3,1,6,8,2,4,7,0],[5,1,3,6,8,2,4,7,0],[4,2,8,6,3,1,5,7,0],[2,4,8,6,3,1,5,7,0],[4,2,6,8,3,1,5,7,0],[2,4,6,8,3,1,5,7,0],[3,6,1,4,8,2,5,7,0],[3,1,6,4,8,2,5,7,0],[2,8,4,6,1,3,5,7,0],[4,2,8,6,1,3,5,7,0],[2,4,8,6,1,3,5,7,0],[2,6,4,8,1,3,5,7,0],[4,2,6,8,1,3,5,7,0],[2,4,6,8,1,3,5,7,0],[2,6,4,1,8,3,5,7,0],[2,4,6,1,8,3,5,7,0],[5,7,3,1,6,4,2,8,0],[4,6,1,3,7,5,2,8,0],[5,3,7,1,4,6,2,8,0],[3,5,7,1,4,6,2,8,0],[6,4,2,7,5,1,3,8,0],[5,7,2,4,6,1,3,8,0],[6,2,4,7,1,5,3,8,0],[2,6,4,1,7,5,3,8,0],[5,2,7,4,1,6,3,8,0],[6,3,1,5,7,2,4,8,0],[6,1,3,5,7,2,4,8,0],[2,7,5,3,1,6,4,8,0],[2,5,7,3,1,6,4,8,0],[3,6,1,4,7,2,5,8,0],[6,2,4,7,1,3,5,8,0],[2,6,4,1,7,3,5,8,0],[4,2,7,5,3,1,6,8,0],[5,3,1,7,4,2,6,8,0],[3,5,1,7,4,2,6,8,0],[4,2,7,5,1,3,6,8,0],[3,1,5,7,2,4,6,8,0]]

Best Answer

Related Solutions

[Math] Does this knot invariant distinguish trefoil chiralities

Related Question