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.
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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
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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
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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
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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
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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
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(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
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(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.)
Best Answer
A "long topological knot" in $\mathbb R^n$ is a topological embedding $f : \mathbb R \to \mathbb R^n$ such that $f(x) = (x,0)$ for all $x \in \mathbb R \setminus (-1,1)$.
Let $K_n$ be the space of all long topological knots in $\mathbb R^n$ with the compact-open topology. Then $K_n$ is contractible. The contraction is given by
$F : [0,1] \times K_n \to K_n$ defined by
$F(t,f)(x) = (1-t)f(\frac{x}{1-t})$ provided $t \in [0,1)$ and $F(1,f)(x) = (x,0)$.
This map, $F$, is sometimes called "The Alexander Trick". See the Wikipedia Alexander Trick page for context.
In response to your edit, perhaps an interesting topology on $K_n$ could be given this way. Given $f \in K_n$ and $\epsilon > 0$ we'll say an $\epsilon$-ball about $f$ consists of all knots $\phi \circ f$ where $\phi : \mathbb R^n \to \mathbb R^n$ is a homeomorphism which agrees with the identity map outside of $D^n$, and such that $|\phi(x)-x|<\epsilon$ for all $x \in \mathbb R^n$. The topology on $K_n$ could be the topology generated by all $\epsilon$-balls about all $f \in K_n$. Presumably this kind of topology has a name?