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While reading this question posted at this link:

i interestingly landed up on this Wikipedia page, and was quite amazed to see the variety of branches opening up. That's how I came to know about the subject Arithmetic Topology. Wikipedia describes this subject as:

  • Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. In the 1960s topological interpretations of class field theory were given by John Tate based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier based on Étale cohomology. In subsequent years, Barry Mazur and Yuri Manin pointed out a series of analogies between prime ideals and knots.

Can anyone explain me as to how Prime Ideals and Knot's are analogous (elementary explanation would be helpful) and why are we interested in studying these analogies.

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Here are some relevant MO questions. –  Zev Chonoles May 13 '11 at 18:11
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3 Answers

up vote 5 down vote accepted

Here's a useful "conceptual analogy dictionary" between 3-manifolds and number rings, excerpted from Morishita: Analogies between knots and primes, 3-manifolds and number rings.

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This is briefly described in Baez's week257 as well as in a post on lieven le bruyn's blog. I do not think there is a particularly elementary explanation here, but as I understand it there are three basic ideas:

  • From the point of view of étale cohomology, $\text{Spec } \mathbb{Z}$ is like a $3$-manifold. I confess I don't really understand what this means, although there is a nice heuristic explanation in the last bit of week257. Apparently it means there is an analogue of Poincaré duality in this setting with $n = 3$.
  • In fact, it is like a $3$-manifold with trivial fundamental group, so is like the $3$-sphere $S^3$.
  • The category of finite extensions of a finite field $\mathbb{F}_q$ looks exactly like the category of finite covers of the circle $S^1$; there's exactly one extension / cover for every positive integer $n$ with automorphism group $\mathbb{Z}/n\mathbb{Z}$. So the map $\text{Spec } \mathbb{F}_p \to \text{Spec } \mathbb{Z}$ is supposed to look like an inclusion of $S^1$ into $S^3$, which is a knot.

As for why we are interested in studying these analogies... I don't know about you, but I find this connection inherently interesting. Analogies of this kind have traditionally been very fruitful for both of the fields they connect in mathematics; if nothing else, it's nice to have a different source of intuition and motivation to make conjectures from.

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I will try to write more later. In the meantime, here is a link to a manuscript of Mazur on this topic. (See the paper Remarks on the Alexander polynomial.)

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Awaiting your answer. –  user9413 May 16 '11 at 18:30
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