# Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena

I am trying to adress Morishita's book Knots and Primes to discover a bit about Arithmetic Topology, but some analogies puzzle me. I know that the correspondence should be addressed with a grain of salt, but some parts of it look so fundamental that I would like to understand them better.

1. In table (3.3) on page 50 of his book, Morishita writes that the longitude, called $$\beta$$, should correspond to a lift of Frobenius and the meridian, called $$\alpha$$, corresponds to a generator of tame inertia (both as elements in the maximal tame quotient of the absolute Galois group of a local field). He calls "longitude" a path going around one hole in the boundary of a tubular neighborhood of the knot and "meridian" the border of a disk which is a "cross-section" of the tube. Were the knot the unknot, this neighborhood would be the full torus $$S^1\times D^2$$: in this case, $$\alpha=\partial D^2$$ and $$\beta=S^1$$. This analogy does not agree with my idea that inertia acts as monodromy, which is "running around holes", but I tried to pursue. Then (page 63, after Theorem 5.1) he describes the analogue of decomposition groups for an unramified knot $$K$$: he says that this group should be generated by "a loop going around $$K$$", which I think is just the image of $$\alpha$$. Then I am completely lost, as I would expect decomposition groups in unramified situations to be generated by Frobenius...
2. In Chapter 11, a tentative analogy with Iwasawa theory is suggested. Nevertheless, it seems to me that something is strange, as typical Iwasawa theory is concerned with very wild ramification, whereas the same table (3.3) on page 50 as before seems to indicate that there is no wild topological inertia. Thus, the Galois group of $$X_\infty/X_K$$, where $$X_K$$ is a knot complement and $$X_\infty$$ is a $$\mathbb{Z}$$-cover of $$X_K$$, looks to me somehow similar to an "infinite tamely-ramified cover" (which has no arithmetic analogue) rather than a $$\mathbb{Z}_p$$-extension. Am I missing something? Along the same lines, he has a small parenthesis between p. 144 and p. 145 where he writes that "[assuming that a $$\mathbb{Z}_p$$-extension be ramified at one prime only, and that this prime be totally ramified] is an assumption analogous to the knot case". Why is it so? Neither the fact that a $$\mathbb{Z}$$-cover must be ramified at only one knot nor the fact that there can't be a small unramified layer below look obvious to me (at least if the base manifold is not $$S^3$$, otherwise $$\pi_1(S^3)=0$$ should say that no unramified extension exist).
• Why does this not agree with the idea that inertia acts as local monodromy, which is running around holds? $D^2$ is the hole, $\partial D^2$ is running around it. – Will Sawin Jul 6 '19 at 15:57
• Why can't "a loop going around $K$" be $\beta$? – Will Sawin Jul 6 '19 at 15:59
• Presumably the assumption that the $\mathbb Z_p$-extension be ramified at one prime only is analogous to the knot case, not the link case. Perhaps he is thinking specifically of knots in $S^3$ there? – Will Sawin Jul 6 '19 at 16:01
• @WillSawin Well, in considering which loop should be the "right" monodromy, I was thinking at $\alpha$ because I have in mind the idea of a family of curves over the punctured disk (in $\mathbb{C}$, say), or over $\mathbb{Z}_p$; but I agree that this is just a feeling. – Filippo Alberto Edoardo Jul 6 '19 at 19:14
• Concerning $\alpha$ vs. $\beta$, remember the relevant Galois groups here are fund'l groups of complements of the neighb'd: for the unknot, for instance, this $\pi_1$ is $\mathbb{Z}$, generated by something which becomes $\alpha$, if I am not mistaken (I am looking at Morishita's ex. 2.6, p. 12). But of course I'd be very happy to be proven wrong and to find again my Frobenius generating the decomposition group! – Filippo Alberto Edoardo Jul 6 '19 at 19:18