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To the editor and the authors:I have a mixed opinion of this paper -- it has many good qualities and should definitely be published eventually, but also has some serious flaws, which do not allow me to recommend it for publication at this time. I'll first give some general comments, then detailed comments further below.
General comments:
The paper is about embeddings of vertices of graphs in the circle. It is a "lower-dimensional version" of spatial and planar embeddings of graphs and intrinsically linked graphs. The results are original and non-trivial (much more than just "homework exercises"). The introduction is well written; but in order to be accessible to undergraduate students not familiar with this field, the introduction needs lots of examples and pictures (it has none right now).
The key definition (of a link in S^0) is not stated clearly. There is a missing definition (triangle-Y exchange). Some of the proofs are incomplete or incorrect. Some terminology is used incorrectly.
I read only the first ten pages (to the end of the proof of Theorem 4.6), then decided to stop due to the above-mentioned problems. It is likely that the statements of the results in the paper are all (or mostly) correct, and only some definitions and proofs need to be fixed. However, I am not inclined to read this paper any further since the errors and gaps and the misuse of terminology make it too time-consuming to follow. The paper needs to be proof-read carefully by the authors or, preferably, by a mathematician with expertise in this field before being resubmitted.
Detailed comments:
Intro: Give the definition of "triangle-Y exchange."
P. 2:
Your definition of "0-sphere" is different from [3]. It is worth mentioning this and explaining why.I think it's a bad idea to denote a 0-sphere as an n-tuple rather than just as a 2-tuple. By definition, (x,y) is a 0-sphere if there is a path from x to y in G. But there may be more than one such path. So the path should not be part of the notation for the 0-sphere, in my opinion. This n-tuple notation confused me when reading Theorem 4.1.
In your definition of a link (a,b) \union (c,d), it is not specified that there must exist DISJOINT paths from a to b and from c to d. But I believe you do want them to be disjoint otherwise in a connected graph any two vertices form a 0-sphere.
P. 3:
I think I've heard that O'Donnol's result in [8] has changed: I think it may be K_{2n,2n+1} instead of {K_2n,2n}. But I may have this all wrong. You should check with the author.
Middle paragraph: "...at most two distinct faces whose boundaries contain all of the vertices." Do you mean "each of whose boundaries" or "union of the boundaries?"
P. 4:
First paragraph: "... it is necessary to show that vertex expansion ..." No, it is not necessary! The set is finite regardless (by [9]).
Proof of Theorem 2.1: It's missing Case 3: (w_1, ..., w_i, v) (i.e., v is an endpoint). This case is different from the other two cases, and is important.
Also, this proof doesn't seem to explicitly have anything to do with the fact that these are THREE-component links. It seems the proof will work for n-links. You should address this point, either way.
You should mention that this proof is the same as the one in [3] (and you're obviously borrowing from [3]) except that there is an additional case to check because the definiton of "link" in this paper is different from that in [3] (not because it has anything to do with 3-links instead of 2-links, as far as I could tell). Be careful with whether the paths must be disjoint or not.
P. 5:
In the proof of Theorem 3.1, define a,b,c,v. Also, don't use the same notation (a,b) for 0-spheres and edges.
" that results from deleting vertex v and " : deleting v from the graph or from the image of the embedding? Be clear. When you delete a vertex from a graph, you also delete all edges incident with it, so the graph may become disconnected, which may change things a lot.
The proof of Theorem 3.1 is incorrect or incomplete:
In Case 1, do you really mean 0-spheres by (a,b), (b,c), and (c,a) or do you mean edges? If 0-spheres (which is what you write), then the proof is wrong, because paths matter; so I assume you mean edges, in which case you must explain why after the triangle-Y exchange the three links are still links (why do there still exist three DISJOINT paths?).
Also, in your definition of 0-sphere, you require SIMPLE paths. This makes the proof of Case 2 more delicate, which you seem to have either missed or chosen to ignore. Why do you require SIMPLE paths in your definition anyway? If you really do want this, then you need more detail and care with the proof of Theorem 3.1.
Finally, this theorem and its proof don't seem to have much to do with THREE-links; I think it should work for n-links as well; you should check this and comment on it.
Question: Does a Y-triangle exchange preserve S^1 linkedness? If you don't know the answer, then I suggest posing it as a question.
Last sentence on this page: "It turns out that it does not preserve this property." Either give a proof or take out this claim.
P. 6:
Proof of Theorem 3.2, Case 2: "Then H' is a minor of G." Should prove this.
The paragraph right before Section 4 is very nice!
P. 7:
I think Theorem 4.1 should be called a lemma.
In the statement of Theorem 4.1, when you say disjoint 0-spheres, that only means disjoint endpoints; do you really mean disjoint paths?
In the proof of Lemma 4.2, you should omit the 4 cases; they are obvious. After the first paragraph, just say "So the lemma follows".
P. 10:
Proof of 4.6: When you "delete" the 0-sphere (a,b), do you delete both vertices or just one? What if one of them (but not both) is involved in one of the three links? It's not clear whether your proof addresses this scenario.
Later in the same proof: "Delete edge (a,b) from this embedding". This has no meaning -- edges are not embedded; only vertices are. What do you really mean? Be careful and precise with the terminology. The proof is incorrect or incomplete.
I got frustrated at this point and stopped reading any further. I know this whole report sounds somewhat harsh; I hope I haven't discouraged you, though! You all have done an awesome job; just check everything again very carefully and fix the errors and gaps and the imprecise terminology. It has great potential.
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