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What is an Artin L-function, conceptually? It must be something really obvious:
The Frobenius homomorphism is to be thought of as a element of the fundamental group and its action via a Galois representation as a holonomy/monodromy around that element.
Hence Artin L-functions are simply (products of) characteristic polynomials of monodromies.
Googling “monodromy characteristic polynomial ” yields… the Alexander polynomial.
Check out this sentence from the introduction of Stallings 87, it is about knots but reads just like the definition of the Artin L-function:
For a fiber surface $T$, the translation of the fibre around the base-space circle determines an element in the mapping-class group of $T$, a homeomorphism $h\colon T \to T$ well defined up to isotopy; this element is called the holonomy of the fiber surface; the Alexander polynomial is the characteristic polynomial of the map the holonomy induces on $H_1(T)$.
This relation/analogy must be know, however Google doesn’t give further hints.
Does this relate to the ’MKR dictionary’ between 3-dimensional topology and number theory? Item (10) on p. 6 of this relates the Alexander polynomial to some character polynomial in $p$.
Mazur’s note Remarks on the Alexander Polynomial might be worth reading too.
Thank you, David, these are really useful links for that general idea of relating knots to primes. I have added pointers to these to the entry Alexander polynomial.
I think the place to go is section 7, ’Alexander-Fox, torsion theory and Iwasawa theory’ of Masanori Morishita, Analogies between knots and primes, 3-manifolds and number fields.
We have a warning at Iwasawa-Tate theory that this dffers from but is related to Iwasawa theory. So I’ll start a stub for the latter: Iwasawa theory. It seems a main conjecture there involves p-adic L-functions.
Re #1, if the Artin L-function is a product over primes, I wonder how close the analogy is. From what I’ve seen Alexander-Fox and Iwasawa sound like they’re working with generic primes.
I am in a deadline problem so I can not get into your references, but as my student MS explained me few times, the holonomy and the polynomial geometrically come about from the study of the Seifert surface of the knot. Tim will know details. See also Fox derivative where I explained the connection of the Alexander polynomial to the Jacobian..
Another MO question: prime-numbers-as-knots-alexander-polynomial. Minhyong Kim says “I would have given a substantial answer to this question if I were not so lazy.”, maybe he should be asked…
The analogies between Alexander polynomial and L-functions and touched upon geometrically in
The same author has other relevant references
It seems in this context, people talk of Iwasawa polynomial instead and the definition is completely analogous (by the way is there a case or an analogue of an Alexander quandle there as well?).
If we can find someone knowledgeable enough, that should be linked with the stub which needs to be expanded for Iwasawa theory.
Alexander quandle and polynomial are defined from finite rank modules over the integral Laurent polynomial ring $\mathbb{Z}[t,t^{-1}]$. For Iwasawa polynomial one uses torsion modules over ring of the formal power series $\mathbb{Z}_p[[t]]$ in one variable over $\mathbb{Z}_p$, cf. Wikipedia, Iwasawa_algebra and our Fox derivative, toward the end. In both cases one looks at the generator of the characteristic ideal. I started Iwasawa algebra just to record a reference.
David,
re #6, the primes: if the analogy with the Selberg zeta here is anything to go by, then the product is supposed to be over generators of the fundamental group. For suitable arithmetic schemes the Frobenius morphisms at the primes provide just that. So this seems to make good sense to me, at this rough level.
Zoran,
re#9, the references: thanks, that’s excellent! Just what I would have hoped for. Will try to look at these references a little later today. Thanks again.
Don’t forget the link between the Jacobian (Alexander matrix) and the ‘abelianised’ form of the crossed complex / chains on the universal cover as that may be what is giving the link to the geometric topology (knot theory).
The first four pages of
(which Zoran had kindly mentiond in #9) nicely present the analogy.
I notice (just for the record, since this relates to previous discussion) that by this analogy – if one were to extent it in $q$ to $q \to 1$ – would not say that $Spec(\mathbb{Z})$ is (real-) 3-dimensional and containing knots, but would say instead that $Spec(\mathbb{Z})$ is analogous to a (real-) 2-dimensional Seifert surface of a knot.
For interpreting what’s going on it seems crucial that the actual Seifert surfaces of course have boundary – the given knot, after all –, which is a rather drastic generalization of the “function field analogy”-story, compared at least what we have discussed so far (and what is usually discussed, unless I am missing something).
Right now I am quite unsure how to best think about what this analogy is supposed to be telling me.
There were variations on the MKR dictionary proposed, e.g., one by Reznikov in Embedded incompressible surfaces and homology of ramified coverings of three-manifolds, Selecta Math. 6(2000) 1–39 where a number field is associated with what he calls a $3\frac{1}{2}$-manifold, that is a closed three-manifold $M$, bounding a four-manifold $N$, such that the map of fundamental groups $\pi_1(M) \to \pi_1(N)$ is surjective.
Urs
Re #13, yes, but my point was that the Alexander polynomial is not a product.
I think I’m pointing to the difference between local zeta function and global zeta function. Wikipedia says the former is also known as ’congruent’, and global ones are products of these, e.g. at Hasse–Weil zeta function.
Sugiyama in remark 3.3 recasts the Alexander polynomial as a certain congruent zeta function, that’s local.
David,
okay, right. And thanks for all the further pointers.
I am beginning to lag behind now with reading. If you haven’t already, could you add a pointer to that remark 3.3 to the entry? I’ll get back to it then as soon as a find the time. This all looks very good.
Okay, began, but quickly ran out of time.
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