Prime numbers as knots: Alexander polynomial A naive and idle number theory question from a topologist (but not a knot theorist):
I have heard it said (and this came up recently at MO) that there is a fruitful analogy between Spec $\mathbb Z$ and the $3$-sphere. I gather that from an etale point of view the former is $3$-dimensional and simply connected; from the same point of view the subschemes Spec $\mathbb Z/p$ are $1$-dimensional and very much like circles; and the Legendre symbols for two odd primes that figure in quadratic reciprocity are said to be analogous to linking numbers of knots. So, prompted by a recent MO question, I started thinking:
The abelianized fundamental group of the complement of Spec $\mathbb Z/p$ (the group of $p$-adic units) is not terribly different from the abelianized fundamental group of a knot complement (an infinite cyclic group). For nontrivial knots, there is a lot more to the fundamental group of a knot complement than its abelianization. The next little bit, the abelianization of the commutator subgroup (or $H_1$ of the infinite cyclic cover) has an action of that infinite cyclic group, and I recall that the Alexander polynomial of the knot may be created out of this action. 
So there must be some analogue of that in number theory, right? Like, some construction involving ideal class groups or idele class groups of $p$-power cyclotomic fields can be interpreted as the Alexander polynomial of a prime number?
 A: This article appears to discuss the relationship between Alexander polynomials in knot theory and Iwasawa polynomials in number theory, although I haven't looked at it in detail. I discovered this paper in This Week's Finds 257, which gives a number of references for this sort of thing.
A: A whole book has recently appeared (Primes and Knots,
Edited by Toshitake Kohno, University of Tokyo, Japan, and Masanori Morishita, Kyushu University, Fukuoka, Japan).  
This volume deals systematically with connections between algebraic number theory and low-dimensional topology. Of particular note are various inspiring interactions between number theory and low-dimensional topology discussed in most papers in this volume. For example, quite interesting are the use of arithmetic methods in knot theory and the use of topological methods in Galois theory. Also, expository papers in both number theory and topology included in the volume can help a wide group of readers to understand both fields as well as the interesting analogies and relations that bring them together.
