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The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was not promoted by SandyGeorgia 03:34, 28 March 2009 [1].

Knot theory

[edit]
Nominator(s): C S (talk)

Substantial editing of this article over the last couple years by me and some other editors have resulted in what I believe to be a well-written and sourced article on knot theory accessible to the "intelligent layman". I think all the FA criteria are satisfied, but I've never nominated an article for FA before, so please be gentle :-) C S (talk) 10:42, 15 March 2009 (UTC)[reply]

Comments by Sasata

Cool! Knot theory. Something I can relate to, having used shoelaces since childhood.

I hope you've been using the right knot to tie them. Even famous knot theorists have been known to make that mistake.
  • The lead is a bit thin, and one-sentence paragraphs are frowned upon, especially in the lead.
  • I expanded and revised the lede. How's it now?
  • There's also a couple of isolated sentences in the history section.
  • I tried to make it flow more, and at least there are no single sentence paragraphs anymore. How's it now?
  • wikilink iconography
  • Done. Actually I didn't like how that sentence read so I redid it by importing some examples instead from the history article.
  • It's difficult for me to believe that no-one bothered applied mathematical theory to knots until the late 19th century. Not saying the statement is wrong, but just implausible to this bystander.
  • Doesn't seem that implausible to me (topology didn't exist as a subject until then) :-). I made a correction though. Gauss had studied knots earlier in the 1830s, but this was something of an isolated event, and his discoveries were rediscovered later by Maxwell, etc. So I put that in. Good catch -- somehow I had put it in the more fuller history of knot theory article but omitted it here.
  • Does the Mobius strip have anything to do with knot theory?
  • Sure. It's a basic topological object and knot theory is topological.
  • (Sossinsky 2002, p. 71-89) ndashes are required for number ranges.
  • Done. query: Should all the references in the reference section have ndashes too?
  • Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not. Would like to know more about this (or at least a reference). I've used spectroscopic methods to determine chirality (the common method), but have never heard of applying knot theory.
  • The Flapan book ref given a sentence later suffices. But I've added an early journal reference to Jon Simon.
  • "Knot theoretic topology may be crucial in the construction of quantum computers, through the model of topological quantum computation." Source?
  • Source is given in the history of knot theory article (and any source for topological quantum computation would suffice too), but I've added it to this article.
  • "Algorithms exist to solve this problem, with the first given by Wolfgang Haken." When?
  • Ok, added. The papers are spread out over a time period so I just said late 1960s.
  • "Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is (Hass 1998)." Why are they time-intensive? Are they solved by hand or computer? The last half of the sentence is also unclear. Is have visions of mathematicians sitting around a table trying to solve a knot problem, half of them arguing ..."you don't understand, it's too hard...". Sounds like it would make a good cartoon.
  • They just are. Asking why is like asking why factoring numbers is hard, Nobody knows why! Some algorithms are partially implemented on computers, but only partially because the algorithms can be complicated. Theoretically it is already clear some algorithms will not work in practice (with the limits on computing resources). In the case of unknotting, some algorithms are fully implemented. In the case of general knot recognition, no full implementation of Haken's algorithm exists. Yes, sitting around the table arguing is a perhaps too accurate picture. It's the same as in any subject where there are many unresolved basic questions.
  • what do you mean "spaghetti with the ends"? Isn't spaghetti just a blob? :-) (so the topologist says while drinking his doughnut and trying to bite a coffee mug...)--C S (talk) 07:56, 16 March 2009 (UTC)[reply]
  • If you think it would be helpful to include any of this, I can. But to take the "factoring numbers" example, this is something that is actually not understood at all with lots of investigative work being done and much unpublished info. I tried to summarize things as best I could.
  • "The special case of recognizing the unknot, called the unknotting problem, is of particular interest." Why?
  • It's just a particularly appealing simple case. Is a loop knotted or not? That's a basic question, and given the complexity of the general question, one can hope to make progress on the easier situation. The Hass ref suffices for this statement, but I added another ref anyway to get a bit of variety (since this statement is more reflective of opinion and sociology).
  • "At each crossing we must indicate..." Should be reworded to take out "we".
  • Ok.
  • "If by following the diagram the knot alternately crosses itself "over" and "under", then the diagram represents a particularly well-studied class of knot, alternating knots." I don't understand, isn't the alternating knot just a consequence of the projection of a regular knot onto a plane? Or is the distinction that the crossing is alternatively "over" and "under" (rather than say, two "unders")?
  • an alternating knot by definition has an alternating diagram, but not every knot has such a diagram, e.g. there are non-alternating knots. In any case, this is something of a digression (originally added by a random passer-by), so I removed it.
  • "A knot invariant is a "quantity" that is the same for equivalent knots." Still don't know what this means; the rest of the short paragraph doesn't help much. Since the whole section is about knot invariants, I think this should be more clearly explained. Having difficulty following the rest of this section.
  • This is a bit of repetition from the lede. Do you prefer the wording from the lede? It talks about the "quantity" being the same for different descriptions of the knot. I added an example to make it clearer. Does that help?

*I'm going to stop here and register an oppose vote. Will check back later to see if its more comprehensible to me. Sasata (talk) 15:39, 15 March 2009 (UTC)[reply]

I'm striking out the oppose vote (want to stay neutral), and will come back again with another read and more comments after the nom has had time to deal with Jakob.scholbach's comments below. Sasata (talk) 21:17, 16 March 2009 (UTC)[reply]

Ok fresh comments, from the beginning. Sasata (talk) 02:08, 16 March 2009 (UTC)[reply]

  • wikilink link (in the lead)
  • Fixed.
  • Is "quantity" the actual word in the literature to represent knots? I associate quantity with "amount", and it seems like "state" would be an appropriate term (similar to the sense of State (physics), quantum state, or Chemical state, but with knots instead of atoms).


  • No, but quantity is not referring to knots. It's referring to knot invariants, which are quantities assigned to knots. More below.
  • In the last paragraph it says "Knots can be considered in other three-dimensional spaces." but no indication is given why someone would want to do this. Perhaps another sentence to mention the practical advantage of conceptualizing knots in n-dimensional space (I'm assuming there is one!).
  • Practical advantage? Mathematicians generalize. This is what they do! I guess I can throw in a sentence like "...to gain further insight, mathematicians generalize..."
  • "...enabling the use of geometry in defining new, powerful knot invariants." In what way are these fancy hyperbolic knots "powerful"? ... Actually, I sort of understand now that I read the next two sentences. However, I'm going to remove the extra spaces around the picture of Tait, so that it looks less like two separate short paragraphs.
  • This seems based on the misconception about quantity above. Since knot invariants are used to distinguish knots, "powerful" here just means they are very effective at distinguishing knots.
  • The closely related theory of tangles have been effectively used..." Perhaps it could be inserted parenthetically (in layman's terms) how tangles differ from knots.
  • Ok. Done.
  • "Knot theoretic topology may be crucial..." Would it be equivalent to use the simpler (and previously used) term "Knot theory"?
  • This linguistic contortion is actually being used for a particular reason. Knots themselves do not show up in topological quantum computation, but the mathematical techniques used and developed by knot theorists do. This may be anal-retentive though...I'll just change it to "knot theory".
  • "When mathematical topologists..." Are there other kinds of topologists?
  • good point.
  • "The idea of knot equivalence is to give a precise definition of when two embeddings should be considered the same." Here the new term "embeddings" is used, but it seems to me to be equivalent to the term "quantity" used above, no?
  • No, again, "quantity" was referring to knot invariant. When reading the article again, I'm very puzzled how it is possible to conclude that "knot invariant" is another term for "knot". I'm wondering whether you simply misread some passages, or whether the misunderstanding is due to the writing being horrible. You realize that I've modified the descriptions of knot invariant in the lede and introduction to the section "knot invariant" since the first time you looked? I'm not sure if you've read them. --C S (talk) 06:59, 16 March 2009 (UTC)[reply]
  • Also, the term embedding was used in the lede, but probably it is not really necessary in this section, so I just removed the term and rewrote the sentence. --C S (talk) 07:54, 16 March 2009 (UTC)[reply]
  • "The basic problem of knot theory, the recognition problem, can thus be stated: given two knots, determine whether they are equivalent or not." This reads to me too much like a math text. How about instead something like: "The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots."
  • Fixed.
  • The two diagrams in this section are awkwardly placed, in that at some fairly typical browser sizes, the upper left corner of the lower picture partially obscurs a word in the lower right hand corner of the text. Suggest placing the pics side by side. See if you like what the following code produces:

{{double image|right|unknots.svg|200|Ochiai unknot.svg|200|(Left) The unknot, and a knot equivalent to it. (Right) It is more difficult to determine whether complex knots such as this are equivalent to the unknot.}}


  • "A small perturbation in the choice of projection..." Suggest changing "perturbation" to "change", and clarify what is meant by "choice of projection".
  • Good points. All that is meant is that changing the direction of the projection slightly will give you a knot diagram. So I just said that instead :-)
  • "...except at isolated times when an "event" or "catastrophe" occurs..."
  • Isolated is referring to the fact that these times are discrete. This paragraph is really a very intuitive description of something considerably more technical. There might be a whole interval of time where events happen, but the point is that you can arrange it so it doesn't happen that way Actually, it suffices to just say "finitely many times"...I hope the new wording is clear.
  • "A close inspection will show that complicated events can be eliminated, leaving only the simplest events which are precisely the Reidemeister moves" Not sure what is meant by the underlined part. Does it mean that events with multiple strands crossing over can be excluded in the analysis, and that two knots are considered equivalent if they are constructed with a related sequence of Reidemeister moves?
  • The Reidemeister moves can be thought of as three types of events: 1) straightening a kink 2) two strands becoming tangent at a point as they pass each other 3) three strands coming together at a point and then moving away. More complicated events would be anything that can happen as you project the knot to a plane but don't have a diagram. So, for example, if more than three strands comes together at a point. Or if three strands become tangent at a point. Etc. I'll try and reword this.
  • Also, to clarify: if two diagrams are connected by Reidemeister moves, then they represent the same knot (because the Reidemeister moves clearly give a movement of the knot that gives the equivalence). It's not as clear that if two diagrams represent the same knot that there are these moves that connect them. That's the point of this analysis.
  • "An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots." Change the sentence structure so it doesn't sound like the knot is trying to perform the action of distinguishing.
  • It doesn't sound like that to me....is this related to the misconception from above that an invariant is a knot?


  • Tech. Review
  • Dabs are not up to speed (based on the checker tool in the toolbox)
    • They need fixing.
Fixed.

Comment

  • I see there are many references already, but some parts still miss them. Anyway, if you would have used the footnotes, you could display also the concrete pages, which would make it much more transparent :) --  LYKANTROP  ✉ 13:15, 18 March 2009 (UTC)[reply]
  • Comments -
  • Is the Collins ref this article from Scientific American? If so, it should be formatted like a journal, not a book.
Otherwise, sources look okay, links checked out with the link checker tool. Ealdgyth - Talk 16:11, 16 March 2009 (UTC)[reply]

Comments Jakob.scholbach (talk) 20:25, 16 March 2009 (UTC)[reply]

I think this is a pretty nice article, well-written in many places, with good emphasis on explanations of "standard" facts etc. However, I have a number of problems: in many places the parlance is not really encyclopedic, which should be easy to fix. Perhaps find a copyeditor? Also, the organization of the article as a whole could use a clearer structure. I feel it would be good to have a section simply called "Knots [and links]", which introduces a couple of basic notions such as knots, links, ambiant space, isotopy, knot complement etc., and also serves as an introductory section. (Notice also that "Knot diagrams" mostly talks about Reidemeister moves. I suggest putting the diagrams part into the intro section). My main concern is comprehensiveness, an FA criterion. As far as I can see, there are quite a bit of applications of k.th., both in mathematics and outside, which deserve a proper treatment. (History section briefly touches upon, but that's not enough). Also, I have the awkward feeling that modern developments get fairly short shrift. (Actually you acknowledge this yourself.) The article does not achieve a clear picture of what modern knot theory is, what are its leading questions, open problems, main techniques etc.

In addition to the above, a couple of more concrete points ('til hyperbolic invariants):

  • "For thousands of years, knots have interested humans" sounds, to me, a bit like a fairy tale.
  • " 20th century - Max Dehn, J. W. Alexander, and others - studied" -- replace hyphens by dashes, see WP:DASH.
  • "In the late 1970s, William Thurston introduced ..." would perhaps be good to cite the paper(s) of Thurson.
  • I may be wrong on that but I think in "and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep" the last comma is not needed.
  • All statements in this paragraph should be sourced.
  • There are several occurrences where multiple references are glued together ( (Adams 2004)(Sossinsky 2002) etc.) I personally think "(Adams 2004, Sossinsky 2002)" would look better. This is possible using the templates.
  • I would spend some time explaining (in this article) ambient isotopies, given the fact that this is quite crucial to understand equivalence of knots.
  • "The basic problem of knot theory" -- perhaps reword to "The fundamental problem"
  • Why is the unknot recognition problem of particular interest?
  • Talking to the reader should be avoided per encyclopedic style ("—think of the knot casting a shadow on the wall")
  • "A small change in the direction" --starting from what?
  • AFAIK, MOS predicts that only the article title is typed boldface.
  • Consider using only one row for the Reidemeister images.
  • "A useful way to visualise" -- useful in what sense? It's a bit unspecific.
  • Are there further ways to depict knots than the 2D projections? There is no clear definition of knot diagrams, but the k. invariants section appears to rely on k. diagrams as a formal notion.
  • If knot polynomial etc. are only the tip of the iceberg, you have to show us the iceberg, at least roughly! (An FA has to be comprehensive).
  • I find it a bit odd that the link notion is placed somewhat oddly into the Knot polyomials section. It feels like this is also a central notion of knot theory, right?
  • looks odd (at my browser). Consider rewriting as L+, L− etc.
  • "Many important knot polynomials can be defined in this way." in what way do you mean? Do you mean that the skein relation is somehow altered? That should be more clear. Also it is not stated (perhaps trivial to knot theorists?) that this procedure describes k.i. for all knots.
  • "The yellow patches indicate where we applied the relation." -- avoid "we" and "one" throughout (MOS).
  • "sneakiness" -- unencylopedic.
  • "These are not equivalent to each other! " -- likewise, remove the "!"
  • "The Jones polynomial " -- highlight both words or none.
  • This may be just me, but I think placing images at the beginning of sections is usually nicer. E.g. in the knot equivalence section the image oddly overlaps with the next section headline etc.
  • I suspect that hyperbolic knots are not knots with some sort of geometric background, but the ones with hyperbolic geometry? If so, reword the first sentence in that section.
  • External links: what is the Kelvin link for? Why is it not a reference? Likewise with the 2nd. External links also need some accessdate.
  • I suggest putting "There are a number of introductions to knot theory. ..." to the References section. Jakob.scholbach (talk) 20:25, 16 March 2009 (UTC)[reply]
(I will not be able to respond to progress before next Monday.) Jakob.scholbach (talk) 23:02, 17 March 2009 (UTC)[reply]

Oppose The history section lists several applications of the theory (statistical physics, chirality, DNA) but the body of this article does not tell a word about it. I think it is very missing and am therefore opposing on the basis of comprehensiveness. I would also like to suggest the authors to merge this article with Knot (mathematics). Vb 11:56, 18 March 2009 (UTC) —Preceding unsigned comment added by 79.233.248.217 (talk) [reply]

  • How do I withdraw the nomination? I find myself agreeing quite a bit with Jakob above. Probably I should expand the article a few more paragraphs on the new homological theories and 4-dimensional invariants (and something on enzymes). Unfortunately, that won't get done any time soon. I would be interested in Sasata's comments on the current revision. I hope most of his objections were addressed. Thanks to everyone for participation -- I think the article has improved from the attention, but unfortunately not to FA standard. --C S (talk) 07:26, 26 March 2009 (UTC)[reply]
I think you don't have to do anything special, just state that you withdraw. SandyGeorgia will then archive the nomination at some point. Jakob.scholbach (talk) 08:20, 26 March 2009 (UTC)[reply]
Please reconsider. This is better than most FAs as it stands, and many of the objections are unimportant MoScruft. Expanions are always welcome, but 4-dimensional invariants should probably be a subarticle anyway. Septentrionalis PMAnderson 22:48, 26 March 2009 (UTC)[reply]
Probably, as I said above, only a few paragraphs are needed, which I could whip up in a few days work, but I would have to dig out some sources and get into the zone. Right now, I'm dealing with other matters, so I can't say when I would be able to get around to working on the article.
The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.
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