I'm trying to address Morishita's book *Nodes and Premiums* to learn a little more about arithmetic topology, but some analogies surprise me. I know that correspondence has to be addressed with a grain of salt, but some parts are so fundamental that I would like to understand them better.

- In the table (3.3) on page 50 of his book, Morishita writes that longitude, called $ beta $, should correspond to a Frobenius lift and meridian, called $ alpha $, corresponds to a taming inertia generator (both as elements of the maximum taming quotient of the absolute Galois group of a local field). It calls "longitude" a path that bypasses a hole in the boundary of a tubular neighborhood of the node and "meridian" the edge of a disc that is a "cross section" of the tube. If the knot was the denouement, this neighborhood would be the complete torus $ S ^ 1 times D ^ 2 $: in that case, $ alpha = partial D ^ 2 $ and $ beta = S ^ 1 $. This analogy does not support my view that inertia acts as a monodromy, which is "turning around holes", but I tried to pursue it. Then (page 63, after Theorem 5.1), he describes the analog of decomposition groups for an uninterpreted node. K $: he says that this group should be generated by "a circulating loop" K $"which I think is just the image of $ alpha $. Then I'm completely lost, because I would expect Frobenius to generate decomposition groups in un-simplified situations …
- In chapter 11, an experimental analogy with the Iwasawa theory is suggested. Nevertheless, it seems to me that something is odd, because the typical Iwasawa theory concerns a very wild branch, whereas the same table (3.3) on page 50 seems to indicate that there is
**no**wild topological inertia. So the Galois group of $ X_ infty / X_K $, or $ X_K $ is a node complement and $ X_ infty $ is a $ mathbb {Z} $-covering $ X_K $, in a sense, resembles an "infinitely thinly branched cover" (which has no arithmetic analog) rather than a $ mathbb {Z} _p $-extension. Am I missing something? In the same sense, he has a small parenthesis between p. 144 and p. 145 where he writes that "[Ensupposantqu'a[assumingthata[ensupposantqu'un[assumingthata$ mathbb {Z} _p $-extension be branched to a first only, and this first be totally ramified]is a hypothesis analogous to the case of knot. "Why is it so? Or the fact that one $ mathbb {Z} $-cover must be branched to a single node, and the fact that there can not be a small, unbranched layer below seems obvious to me (at least if the basic variety is not $ S ^ 3 $, other $ pi_1 (S ^ 3) = $ 0 should say there is no unassembled extension).