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1. Tried to make some improvements. Gave a formal definition of a link diagram (surprisingly difficult to find a fully rigorous one in the literature). Reorganised some of the earlier content.

Lacks a discussion of how to obtain a link from a link diagram.

2. Hi Richard,

• A graph equipped with a planar embedding is sometimes called a “plane graph” rather than a “planar graph” (I like to make this distinction).
• I guess that the first special case of Definition 2.2 is because you don’t want to consider the “vertex-less loop” as an abstract graph? That’s fine…I just note that there’s some related discussion at Remark 6.2 of this paper by Joachim Kock, and he cites some papers giving other ways of dealing with vertex-less loops (but I haven’t read them).
• Another definition (I think very common) of link diagram is as a 4-valent plane graph equipped with the additional data of a bit of information at each vertex, indicating whether it represents an overcrossing or an undercrossing. I believe this is essentially equivalent to the data in Definition 2.1, but maybe I’m missing something? (To be more precise, these are equivalent assuming the plane graph is also checkerboard colored, but such a coloring exists and is almost uniquely determined by the embedding, up to swapping black with white.)
• CommentRowNumber4.
• CommentAuthorRichard Williamson
• CommentTimeAug 26th 2018
• (edited Aug 26th 2018)

Hi Noam, thanks very much! Discussion like this is what I like best about the nLab!

1) I have changed to plane graph now, thanks!

2) Thanks for the reference!

you don’t want to consider the “vertex-less loop” as an abstract graph?

Yes, exactly. But no problem if you or anybody else would prefer to allow it, please feel free to go ahead and edit the entry accordingly.

3) I actually did not know how to make that definition rigorous! I.e. I did not how exactly to rigorously say what the extra information at each vertex is. Do you mean that literally can just decorate each vertex with an element of a two-element set? Ah, yes, maybe I see why this is OK; your remark about checkerboard colouring is very helpful. What I would suggest is that we keep the ’main’ definition as it is, because I think it is simpler and more direct, but it would be great if we could a remark about the other definition together with the explanation of why it is reasonable by means of checkerboard colouring to the entry. Would you like to do that?

3. you don’t want to consider the “vertex-less loop” as an abstract graph?

Yes, exactly. But no problem if you or anybody else would prefer to allow it, please feel free to go ahead and edit the entry accordingly.

My feeling is that if one wants to convey the intuitive meaning of knot/link diagrams, it is better to count the vertex-less loop as an abstract graph (as in many of the informal presentations that I’ve seen) rather than treating it as a special case. On the other hand, if one is in a situation where a fully formal/combinatorial definition is needed, then it may be that there is other structure present that avoids the need for dealing with the vertex-less loop directly. For example, I can imagine that in formal contexts one may actually want to work with pointed links, where each component knot contains a marked point. The shadow of such a pointed link corresponds to a plane graph with 4-valent and 2-valent nodes, and there is no possibility for generating a vertex-less loop (in particular, the shadow of the pointed unknot is the graph with one vertex and one loop edge).

[…] but it would be great if we could a remark about the other definition together with the explanation of why it is reasonable by means of checkerboard colouring to the entry. Would you like to do that?

I don’t have time to do that right now, but maybe eventually? I was already using that informal definition over at virtual knot theory.

• CommentRowNumber6.
• CommentAuthorRichard Williamson
• CommentTimeAug 29th 2018
• (edited Aug 29th 2018)

Thanks for the thoughts, Noam, lots of good points there.

My feeling is that if one wants to convey the intuitive meaning of knot/link diagrams, it is better to count the vertex-less loop as an abstract graph (as in many of the informal presentations that I’ve seen) rather than treating it as a special case.

I agree that it would be more elegant, but I’m not sure that I fully agree yet that it is more intuitive. I tend to think of the 4-valency condition as convenient in its brevity, but I often actually think of, and occasionally even have a need for, subdividing the edges as being allowed, so that one has 2-valent nodes as well, as you were getting at at the end of your first comments, in which case the unknot can be treated in the same way as everything else. So I suppose the definition as it is now is kind of the middle one of (at least) three variations. I’m not at all against allowing a vertex-less graph though. Either way, we should add a remark to the page summarising our discussion of this point!