Talk:Knot theory/Archive 1

To do list
I moved the following todo list from the article (where it doesn't belong) to this Talk page.

Still to come: &mdash;Herbee 13:17, 16 May 2004 (UTC)
 * Gauß diagrams
 * Signed Graph representation of knots
 * History of knot theory, including resurgence since Jones polynomial
 * Maybe try to explain Witten's connection between knots and quantum gravity!

broken link
— Preceding unsigned comment added by Ruud Koot (talk • contribs) 03:58, 2 June 2005 (UTC)

List of knot theory topics
Please help complete the list of knot theory topics by adding relevant articles on knots, braids, links, etc. Michael Hardy 20:10, 9 June 2005 (UTC)


 * No mention has been made of knots and periodic transformations - especially of the p.A. Smith conjecture that the only curve that can be the fixed point set of a (smooth or piece-wise linear) periodic transformation of the 3-sphere is the unknot.  Chuck 22:44, 15 December 2005 (UTC)

You/Me
I'm not comfortable with this article using first and third person ('you'/'me'). The Wiki style manual isn't so keen on it either. Is it all right with everyone if I dive in and do my best to avoid the 'bad' pronouns? Spamguy 22:13, 4 June 2005 (UTC)


 * Personally, I don' think this article is written in a very encyclopedic fashion. It's more like a teacher giving a lesson to a student. I wouldn't mind having a some of the sentances restructured and the personal pronouns taken out. In fact, I could do it sometime in the near future, if that's all right with everyone. Reader12 00:30, 30 April 2006 (UTC)


 * I would be very cautious in how you change things. As far as the "teacher giving a lesson" thing goes, keep in mind that while probably some of the pronoun usage could be favorably excised from this article, good mathematical writing style, in even very formal contexts, often addresses the reader, e.g. "consider...", "suppose...", "Take a knot and form...".  The reason is that sometime a procedural description can improve clarity.  For example, while it is possible to describe the knot sum in a less procedural manner, it would be less clear.  Also, sometimes such a passage that says "you" is actually giving a constructive procedure (in a sense).  For example: "You can unknot any circle in four dimensions. There are two steps to this. First, 'push' the circle into a 3-dimensional subspace. This is the hard, technical part which we will skip. Now imagine temperature to be a fourth dimension to the 3-dimensional space. Then you could make one section of a line cross through the other by simply warming it with your fingers."


 * Here's one way to rewrite it to lose some pronouns:"Any knot in four dimensions is actually unknotted. The proof consists of two parts. First, 'push' the knot into a 3-dimensional subspace.  This is the hard, technical part which we omit.  Now imagine temperature to be a fourth dimension to the 3-dimensional space.  The knot is in a 3-dimensional subspace, so is the same temperature everywhere.  Two nearby segments of the knot can be passed through each other by gradually heating up one segment, moving it through the other, and then letting it cool back down.  Note that this relies on the fact that parts of the knot of different temperature are allowed to intersect."


 * That's not a perfect paragraph, by any means, and needs work still... Anyway, here we've gotten rid of some pronouns. But we still have a "we" and plenty of what could be considered "giving a lession" or at least instructions.  However, it's perfectly acceptable in standard mathematical writing to allow this.  --C S (Talk) 03:36, 30 April 2006 (UTC)


 * For some reason "we" is more acceptable. Compare: "We can unknot any circle in four dimensions. There are two steps to this. First, we 'push' the circle into a 3-dimensional subspace. (This is the hard, technical part, here omitted.) Now we imagine temperature to be a fourth dimension to the 3-dimensional space; then we can make one section of a line cross through the other by simply 'warming it with our fingers'."
 * Also not ideal, but "you" has been excised. My personal style guide is to use "we" to mean "the community of mathematicians", or (sometimes, more intimately) the author(s) and reader(s). In Wikipedia articles, "I" can never be used; in mathematical papers, it may rarely be the perfect choice (meaning the sole author). --KSmrqT 08:09, 1 May 2006 (UTC)


 * Addendum: One solid motivation for using "we" is to substitute active voice for passive voice, to increase the readability, power, and appeal of the sentences. --KSmrqT 08:17, 1 May 2006 (UTC)


 * My personal opinion is that personal pronouns should not be used too heavily; articles shouldn't sound like a conversation between two people, but some usage is acceptable. The two paragraphs that Chan lists above both look good to me. -lethe talk [ +] 05:06, 1 May 2006 (UTC)


 * I'm not a big fan of we. One of my profs at university was quite insistant on removing any instance of we in papers we were writing. He prefered to reserve the we for instances where we were expressing an opinion. For the most part I agree with him. The point about active voice is good, although I think active voice can be acheived without personal pronouns. --Salix alba (talk) 09:32, 1 May 2006 (UTC)


 * I quite like to use we and it is widely used in maths papers and approved of in mathematical style guides. However, here we are writing an encyclopaedia and Charles made the valid point that people not familiar with the mathematical literature will find the mathematicians' use of we rather odd. This holds especially for articles that will be read by non-mathematicians, and this article probably belongs to this group. So, when writing for Wikipedia, I try to use we only sparingly, though I find it hard not to use it at all and still write in a style that I'm happy with.
 * I think you should be avoided on Wikipedia. It can often be replaced by one (though one should also not be overused). The imperative voice is fine, if used in moderation.
 * More important, in my opinion, are other bits that make the article rather informal. I think that "warming with (y)our fingers" (in the fragment quoted above) and "Creating a knot is easy" (in the article itself) should go.
 * On the other hand, Wikipedians should be given some lattitude to write in their personal style. Too much reliance on the style guides may lead to a bland writing style. -- Jitse Niesen (talk) 13:19, 1 May 2006 (UTC)


 * See the discussion at http://meta.wikimedia.org/wiki/Reading_level, particularly the comments about math articles. I think it is a big mistake to make language more formal when doing so makes the article less accessible to beginners. Judicious use of personal pronouns often make it easier to understand what is being said and they should not be edited out just to make the article seem more "serious." --agr 13:58, 1 May 2006 (UTC)


 * Well - Never gotten that kind of response before. I'm not an expert, so - I like "we" and I like the first paragraph mentioned by Chan. I see how using a more personal style makes it easier to understand. I also agree with Jitse when he says that opinionated phrases such as "Creating a knot is easy" needs to go. A little variety is good too. Unfortunately, I now have no time to rewrite this, so I'll watch it when I can and participate as much as possible.--Reader12 21:40, 1 May 2006 (UTC)


 * There is a lot of response probably because this conversation was pointed out on Wikipedia talk:WikiProject Mathematics, which a lot of people follow. -lethe talk [ +] 23:44, 1 May 2006 (UTC)

Adding knots
Hi, I thought I would mention that I've greatly expanded this section with pictures and explanation in Connected sum, which has inadvertently led to duplication. I'm not quite sure whether I should delete the section "Adding knots" since connected sum is a better place for this, or we should just keep duplicating. Two comments: it would be good to incorporate some of the changes in wording of Rick Norwood to the connected sum article. Also, I've commented there that just cutting and tying ends together is not well-defined (leads to different knots); perhaps someone (or me, if I have time) should thrown in an example of this. As it stands, the "simple explanation" is misleading and should only be mentioned to point out that it's not well-defined. --C S 01:17, 14 November 2005 (UTC)


 * On second glance, while equivalent to what I originally wrote, I don't think saying that the other pair of sides of the rectangle is disjoint from the knots is as clear as just saying the whole rectangle should not touch either knot except along one pair of sides. Just my opinion.  --C S 01:34, 14 November 2005 (UTC)


 * I'll add that. Rick Norwood 14:00, 14 November 2005 (UTC)


 * Comment by someone else (not Rick): Re unknots: Need to quote the most famous question in knot theory: "When is a knot not a knot?"   : ) — Preceding unsigned comment added by 65.217.188.20  (talk) 12:30, 18 July 2006 (UTC)

Rewrite to cover the subject better?
So far this article has mainly been a kind of elementary introduction to knot theory rather than an article about knot theory per se. I was thinking of rewriting it as it is probably an article that stands a good chance of being a featured article; however, I've run into some difficulties in conceiving of how to approach this. Should the article, for example, in the intro, describe what the subject is about? The current lead basically just gives a quick description of the basic notion of mathematical knot and equivalence under ambient isotopy. --C S (Talk) 04:45, 10 October 2006 (UTC)


 * I think the invarients need to be covered in more depth, the study of knot invatients seems to be the main theoretical focus and where most of the advanced maths comes in. --Salix alba (talk) 13:53, 10 October 2006 (UTC)

citation for Haken's recognition algorithm
Someone asked for a citation that there is an algorithm (several now) for an algorithm to decide if two knots are the same. I find it tricky to use the citation templates and ref tags, so here is the info: Haken outlines an algorithm in the 60s, and various people filled in the details, with Hemion providing the last big chunk. A ref for this is Hemion's book: The classification of knots and $3$-dimensional spaces. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. ii+163 pp. ISBN: 0-19-859697-9. Also the citation to Haken and Hemion is given in section 7 of an article: Hass, Joel, Algorithms for recognizing knots and $3$-manifolds. (Special issues on knot theory and its applications). Chaos Solitons Fractals 9 (1998), no. 4-5, 569--581.arXiv link. The paper (same section) also gives an algorithm based on geometrization. --C S (Talk) 18:59, 10 October 2006 (UTC)


 * I removed the cn tag and added a link to the unknotting article and another link i found. I think the story needs to be explained better here, but his should help for now. --agr 02:12, 11 October 2006 (UTC)


 * I'm going to remove your links and just add a ref to the paper I linked above. The unknotting problem is a very special case of the more general recognition problem which is also algorithmically decideable.  Also the link to the blog is just a bad idea.  The blog post is a confused discussion of the erroneous MathWorld snippet.  I'd spend some time giving a better description of these issues and format the refs, but I'm planning on rewriting the whole article anyway (got started earlier today).  --C S (Talk) 06:10, 11 October 2006 (UTC)

Does the image of the Reidemeister moves appear below the text for everyone?
The problem seems to be with the space required by the vertically arranged, two images back up in Knot Equivalence. --Eddie | Talk 21:51, 16 May 2007 (UTC)

Definition of knot equivalence
The article says, "Two mathematical knots are considered equivalent if one can be transformed into the other via a continuous deformation (known as an isotopy)". If the knot is the map from the circle to R3, then any knot is isotopic to the unknot: you simply pull on the "ends" making the knot smaller and smaller until it vanishes. --Davidjmarcus 04:21, 29 May 2007 (UTC)


 * The lede section is written in a somewhat looser fashion for accessibility, even more so than the rest of the article. The main body of the article inserts the adverb "smoothly" which avoids the issue you describe.  I will also add "smooth" to this introductory sentence to avoid further confusion (although it would seem to be one of those things where people who know enough to notice the problem know enough to fix it...). --C S (Talk) 07:15, 29 May 2007 (UTC)


 * Actually, on second thought, given the discussion later, probably what was meant in the lede is an ambient isotopy, which also avoids the problem. --C S (Talk) 07:28, 29 May 2007 (UTC)


 * One really needs to specify ambient isotopy (as I have already done in the article) instead of smooth (imbedded) isotopy. Otherwise, polygonal knots and smooth knots cannot be equivalent, even though there are knotted hexagons (equivalent to trefoils). -- Chuck 04:59, 30 May 2007 (UTC)


 * There is a looser sense in which PL and smooth knots are equivalent. There is a 'smoothing map' from the space of PL knots to the space of smooth knots given by systematically rounding the corners. One can show this map sends equivalent knots (in the PL sense) to equivalent knots (in the smooth sense) and is a bijection between equivalence classes of PL knots and smooth knots.  The 'inverse', a map from smooth to PL knots is easier to construct as one just takes PL approximations. Rybu 15:58, 8 July 2007 (UTC)

Possible mistake
The article says:

"Modern knot theory has extended the concept of a knot to higher dimensions."

It is my understanding that knots are only possible in exactly 3 spatial dimensions. Not 2, and not 4. -76.209.63.170 10:10, 17 November 2006 (UTC)


 * You are correct about ordinary knots tied in a string, they easily untie in higher dimensions, but one can knot a 2-sphere in four dimensions. In general an n-sphere can be knotted in n+2 dimensions. This is explained further in the article.--agr 12:12, 17 November 2006 (UTC)


 * Moreover, in the world of smooth manifolds, one can knot, for example, a 3-sphere in 6 dimensions. -- Chuck 23:19, 4 January 2007 (UTC)


 * That's news to me. I thought codimension two was a requirement.  Reference? Rick Norwood 16:09, 5 January 2007 (UTC)


 * Codimension two is only a requirement for the PL category (result of Zeeman). These facts are mentioned in the article, which currently gives some names as references.  For the smooth knotting result, there is a paper by Haefliger called Knotting (4k-1)-spheres in the 6k-sphere; this shows that there are such smooth knots for all k.   These references and further explication will be included in my rewrite that I am working on.  --C S (Talk) 19:34, 11 January 2007 (UTC)


 * I have heard that the higher dimentional knots can be considered as fractal expansions of the 3 dimentional knots. Does anyone else know the source of this idea?--Scorpion451 18:04, 5 July 2007 (UTC)


 * I've done some work on high-dimensional knots and have read almost everything I can find on the subject and have never heard a comment like that before. I'm not sure what a 'fractal expansion' of a knot might be. --Rybu 15:52, 8 July 2007 (UTC)


 * I'm not sure that "fractal expansions" is the proper name for the concept. The idea I'm reffering to was presented as that a trifold equivalent in higher dimentions would be constructed of interfolded trifolds, much as a hypercube is constructed of cubes extending into the higher dimentional space. I am not familiar with the mathematical language used in n-dimentional geometry, so please forgive me if I am being confusing. I should also mention that the article was related to the theory of manifolds, and their stability, specifically how a knot in 3-space could be deconstructed into a torus in 4-space, so the knots had to have a different structure. As I said, I am trying to figure out where this idea came from myself.--Scorpion451 19:57, 8 July 2007 (UTC)

remove some 'see also'(s)
The links to Khipu and Knotane don't extend any of the mathematical ideas listed here. I think they should be removed. AlfredR 21:20, 15 August 2007 (UTC)

Proof there are no 1-knots in 4D space
Does anyone have a reference for the rigorous proof that there are no knots in four space? —Preceding unsigned comment added by 128.100.68.131 (talk) 22:48, 14 October 2007 (UTC)


 * I think Colin Adams addresses this in "The Knot Book". In all textbooks I've seen, this appears as an exercise (if it is addressed at all).  So a rigorous proof is not too hard to come up with.  VectorPosse 00:02, 15 October 2007 (UTC)


 * The observation that any two embeddings of the circle in R^4 are isotopic is pretty old. The first person to write out formal details of such a proof would likely be Hassler Whitney.  The result can be viewed as a consequence of his "Weak Embedding Theorem" which is actually an approximation theorem if you mine it for all it's worth.  I put a sketch of Whitney's original proof up on the sci.math.research mailing list way back... Here is a link to my post: http://groups.google.com/group/sci.math.research/browse_thread/thread/9e7f321be4ac7ed3/7b3438573fe397cb?lnk=st&q=ryan+budney+embedding#7b3438573fe397cb  Whitney's first paper on manifold theory would be the one you're interested in.  It's okay to read, but a much friendlier modern approach would be the Differential Topology book by Guillemin and Pollack.  BTW, here is the Wikipedia entry for the embedding theorem: http://en.wikipedia.org/wiki/Whitney_embedding_theorem Rybu  —Preceding comment was added at 16:49, 17 October 2007 (UTC)


 * Prztytycki claims that Klein originally gave the proof:  I found Klein's original article, but because my German sucks, I can't tell where the proof is or what is actually proven. VectorPosse 19:09, 17 October 2007 (UTC)


 * Ah, that sounds about right. Whitney's proof is much more general than just the circle in R^4 -- it proves that there is only one embedding of an arbitrary compact n-manifold (up to isotopy) in R^m provided m>2n+1. — Preceding unsigned comment added by Rybu (talk • contribs) 21:03, 17 October 2007 (UTC)


 * I like the previous answer, but I thought I would offer a very elementary proof outline, which can be filled in with a good imagination of how Euclidean geometry works in 4 dimensions. I think sometimes people asking for a "rigorous" proof or reference have the idea that this kind of thing is complicated or the "real" proof involves some subtle detail that is difficult to explain.  So hopefully this will show people otherwise.  It's just a bit much to include details like this in an overview article.


 * An important thing to note first is we need to specify whether the knot is polygonal (piecewise linear) or smooth (so we can use calculus). If the knot is just a topological embedding, the result is no longer true, i.e. there are nontrivial "wild" embeddings of the circle in 4D Euclidean space.  It's simplest to use polygons so I will assume the knot K is polygonal.  What we want to do is find a direction to orthogonally project the knot so that it ends up in a 3D hyperplane of the 4D space.  Once we have that, as the article explains, we have a knot in 3D but we have an extra degree of freedom (due to the 4th dimension), so we can effectively cross the knot through itself to unknot it using this freedom.


 * By the way the part I'm about to explain is not explained in The Knot Book, I think, not even in the revised edition. I'm also going to be highly redundant, going case by case instead of all at once, as to hopefully help the reader visualize what's going on.


 * Recall we want to orthogonally push K into a 3D hyperplane. But we want to do this without creating self-intersections.  Start by picking an arbitrary direction.  That corresponds to a family of 3-planes (hyperplanes) all parallel to each other.  We might as well just pick one that doesn't touch K.  Call it E.  Look at all the lines perpendicular to E.  Each line intersects E in a single point.  If K intersects each line in at most one point, then we can simply push K along these lines into E and we are done!  obviously we can't expect to be this lucky and so probably some lines will intersect K in multiple points.  Here's where the polygonal nature of K comes in.  K has finitely many edges.  If an edge of K is contained  in one of these lines, clearly we could have avoided this from the start.  Each edge of K gives a direction and since there are finitely many, we just avoid this finitely many choices for the direction (and thus E).  So we can suppose that no edge of K is contained in a perpendicular line to E.


 * Suppose two edges of K project on top of each other under the orthogonal projection.  These two edges determine a 2-plane.  There are only finitely many 2-planes that come from K in this way.  Each plane gives us an infinite number of directions we need to avoid, but still we can say that the number of such directions coming from finitely many planes is not dense in the space of all directions.  To see this, consider that the space of all directions as a 3-sphere centered at the origin.  Then each 2-plane of directions has intersection with the 3-sphere consisting of a circle and so just pick a point in the 3-sphere that avoids these finitely many circles.


 * Suppose two edges intersect each other under projection but not on top of each other. This means that they are skew and determine a 3-plane.  Again, similar to the method before, we know this gives only finitely many 3-planes.  Each 3-plane's worth of directions in the total 3-sphere of directions will be two-dimensional.  We can simply pick a direction that avoids this two dimensional subset.


 * Once you pick a direction to orthogonally project by avoiding all these bad directions, the resulting projection will keep the knot embedded (no self-intersection). --C S 08:13, 5 November 2007 (UTC)

New To Advanced Math
Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as knot theory, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno — Preceding unsigned comment added by 66.82.9.82 (talk) 21:12, 26 January 2006 (UTC)


 * Wikipedia isn't meant as a replacement for or a supplement to textbooks. Just like a paper encyclopaedia would be insanely long if it picked up the role of textbook by including problems, solutions, and workthroughs, Wikipedia would be a mess.


 * My best recommendation is to read real books if you're interested. I strongly and highly recommend Colin Adams' The Knot Book if you want to learn about knot theory. It's part textbook (my class on knot theory used it as one), part light reading, and it goes much, much deeper than this article ever will. It has problems to work through as well. Once you read it, knot theory is actually pretty fun! Best of luck, Spamguy 23:07, 26 January 2006 (UTC)


 * This article stinks. Oooh I'm math wiz read how I confound you with my inscrutable lingo....  If the article can't convey the subject then why not dumb it down so it doesn't scare people off?  How about encouraging interest in mathematics instead of hindering it?  —Preceding unsigned comment added by 134.193.128.62 (talk) 00:51, 8 November 2007 (UTC)

knot theory rewrite
I started a big rewrite of the article at User:C S/todo/draft7. Comments are appreciated. When I have something satisfactory, I will replace the article with the rewrite and then people can do as they wish. I hope I don't step on any toes, but it's clear the article could use major changes, and people haven't really significantly modified the article in a while. --C S (Talk) 10:32, 10 December 2006 (UTC)


 * Ok, I have done the deed and replaced the article with my rewrite. There were no substantial edits in between versions I started from and replaced; I included a couple interwiki links that have been added.  Two obvious things are: 1) Someone suggested including a small knot table 2) the colors on the two pictures of the Borromean rings (one from "inside" the link complement) do not match.  --C S (Talk) 07:26, 19 March 2007 (UTC)


 * I've added a picture of a knot table in the tabulating knots section. I'm pretty sure it's all correct but I wouldn't mind having someone else double check.  J kasd  16:55, 25 April 2008 (UTC)


 * Nice job. By the way, #2 was taken care of too, in case anyone is wondering.  --C S (talk) 20:14, 1 May 2008 (UTC)

History
The "spiritual inspiration" aspect of knot theory was uncited and unsupported, I toned it down to mentioning the use of knots in fashion and iconography which seems a reasonable statement. If anyone, anywhere has in fact used knot theory or knots in general to gain "spiritual insight into the workings of the universe" in a relevant fashion then perhaps the editor could cite such and give an example. —Preceding unsigned comment added by 220.233.42.204 (talk) 02:15, 24 December 2008 (UTC)

Featured article candidate
I think this article stands a good chance of becoming a featured article, but I don't feel that I'm enough of an expert or a major contributor to nominate the article myself without some consensus. Any comments would be appreciated. If any pictures are missing, I might be able to make some.  J kasd  00:20, 30 April 2008 (UTC)


 * It might be a good idea, but there have been some issues with putting math articles up for featured article candidate. I'll ask WikiProject Mathematics first (and also to give more people heads up on this), since if there's a good reason not to put it up for FA, I'm sure someone will mention it.  --C S (talk) 20:24, 1 May 2008 (UTC)


 * Suggestions from the WikiProject discussion:


 * History: create other articles such as History of knot theory. This will allow more extensive coverage of the mathematics and also be less intimidating for readers of the knot theory article.  Possibly some lede material can be shifted to the summary paragraph(s) of the history subsection.  Include some historical pictures, say, of Tait, Thomson, etc.  A pic of a page from the Book of Kells might be nice too.


 * Referencing: more references. Possibly we should drop Harvard referencing, since that is not so fashionable for FA reviewers.  Some of the more "disputable" (by reviewers) statements should be removed or reworked.  Sossinsky was included by someone as a ref for several of the history paragraphs but actually I used mainly Silver's article for that.  The referencing for the historical section probably needs to be reworked anyway.


 * Knot tabulation: Originally the idea was to create a separate article for this, but was eventually dropped. So now seems a good time to revisit this.  The history of tabulation could be considerably expanded and detailed in such an article, and the knot theory article would only list some very brief history and be more about the mathematics.  One good reference that was not noticed not when this section was first created is Jim Hoste's survey of knot tabulation:
 * Jim Hoste, The enumeration and classification of knots and links, Handbook of Knot Theory, W. Menasco and M. Thistlethwaite, eds., Elsevier (2005) 209--232. PDF available (this has a pretty thorough overview of tabulation history with references)


 * Lede length: after reworking, this may need to be shortened to be more of the average lede length for FAs.


 * Further math to put in:


 * Something more modern like Khovanov homology might be fun to put in. A calculation for the trefoil should be simple to include, although space may be a concern.
 * Dror Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebraic and Geometric Topology 2 (2002) 337–370. arxiv link (this has a pretty elementary accounting of Khovanov homology, has a nice computation of the homology of the trefoil
 * In mentioning higher dimensional knots, it could be advantageous to describe things like slice genus or ribbon knots
 * Would it be too much to add topics like Legendrian knots?
 * And what about physical knot theory? Things like knot energies.  This could make for a good expansion of the science-related material.
 * The science-type applications can be varied and scattered. More sources are needed besides the Flapan book.
 * --C S (talk) 05:41, 26 May 2008 (UTC)


 * I agree with a separate article on knot tabulation. I'd also like to see a separate article on knots in higher dimensions, incorporating the material on that topic in knot (mathematics). I think we need to rationalize having separate articles knot (mathematics) and knot theory. I'd suggest doing this by aiming knot (mathematics) at lay readers and keeping knot theory at about its current level of technicality, maybe with some edits to make the lede a bit more accessible.--agr (talk) 23:44, 27 May 2008 (UTC)


 * There actually is a real need to have a separate article on mathematical knots. There are many different notions of mathematical knots, while knot theory actually in almost all cases refers to the study of the maps of circles into the 3-sphere.  Only a few of those notions are currently described, but there are more (including a kind where the ends don't have to be tied).  So I think that knot (mathematics) should be reworked to be more of a disambiguation page to explain the different ways the term "knot" is used in mathematics.  While I agree that higher dimensional stuff could fruitfully go into another article, I think it really needs its own article with a title like knotted sphere or knot theory (higher dimensions).  --C S (talk) 07:34, 28 May 2008 (UTC)


 * Just FYI. The Legendrian knot article is a depressing stub.  When (and if) I finish my dissertation, I plan to turn this into a proper A-class article.  It would be best left there, although some mention could be made of it here if only to link to it.  It might be mentioned alongside other kinds of knot theory, like virtual knot theory, for example.  VectorPosse (talk) 05:29, 28 May 2008 (UTC)


 * Thanks for the FYI, but note the edit history (I made it one sentence longer a while ago, hooray!). Anyway, I don't doubt you will finish the dissertation, but the way these things work...let's just say I hope you come back to that article one day. :-)  --C S (talk) 07:34, 28 May 2008 (UTC)


 * A damned good sentence at that! :)  I only say depressing because it depresses me that I can't pursue this and other articles on my to-do list.  Alas...  Best of luck in the FA process! VectorPosse (talk) 09:02, 28 May 2008 (UTC)


 * Back to the question of what knot (mathematics) should contain. How does  a reader know which article to read?  I think the knot theory article currently serves well as a disambiguation page, with summaries of each topic and a link to a main article.  Any additional topics should have a mention there. I have no problem with mention other notions of knot in the knot (mathematics) article, but it should have a distinct identity, or be a redirect to knot theory.--agr (talk) 17:09, 28 May 2008 (UTC)


 * I think what you suggest works. Some of the other defintions (like the physical ones) I think could just be mentioned with links to other new articles.  But I don't know what to do about knot (mathematics).  Not all of the info could be shifted over favorably to knot theory.  For example, the paragraph on the use of "knot" as embeddings of submanifolds and so forth.  --C S (talk) 06:12, 29 May 2008 (UTC)


 * Just to clarify, I believe you are suggesting something along the lines of this old discussion about graph (mathematics) versus graph theory. So knot (mathematics) would be a layman's introduction to the basic object, and also some other more general definitions to help disambig links.  Knot theory would then be the umbrella article for the different subarticles (including the one on higher dimensional knot theory).  --C S (talk) 00:14, 30 May 2008 (UTC)


 * Yes, that is what I am suggesting. I'm open to other approaches, but I feel we should agree on an approach before we do a lot of editing. The graph theory discussion is instructive and seems directly applicable here. I suggest looking at the two articles graph theory and graph (mathematics). I'm not thrilled with either of them. The are both dry and skeletal and it's not as clear as I'd like what belongs where. I hope we don't end up with something similar. One a side matter, I think the natural place for the submanifold stuff and the like would be Knots in higher dimensions.--agr (talk) 03:31, 30 May 2008 (UTC)


 * I don't see any objections to the proposal. I note that group (mathematics) and group theory seem to be turning out this way too.  Rybu who considerably expanded knot (mathematics) just edited this article so presumably he has seen this discussion and has no objections.  People might as well move ahead and implement this.  --C S (talk) 06:56, 1 June 2008 (UTC)


 * It's been a while, and I appreciate all the effort that has gone into this so far. I still think that this article is close to being a feature articled candidate. One thing I still think that it is lacking content wise, is some sort of explanation on why the Reidemeister moves are sufficent. Other than this, I think the article is very complete.  J kasd  10:22, 22 February 2009 (UTC)

6 billion knots!!
Article says over 6 billion knots have been catalogued since the 19th Century. That's one knot for everyone on planet Earth! Could that really be true? I mean it has to be coz it's in WP, but I'm astounded. Were most of these knot formats churned out by some computer working by itself? These are knotty problems Myles325a (talk) 05:28, 5 May 2009 (UTC)


 * Yup. The article has the link to the Hoste paper.  These are all the prime knots which have alternating diagrams with 22 crossings or less.   I'm not sure if the table is available or not, as it's got to be a huge file.  The software 'knotscape' has all prime knots up to 16 crossings in their knot diagrams, and this is readily available.  The Hoste paper does not deal with non-alternating knots -- for that you're restricted to the knotscape 16-crossing table.  Rybu (talk) 08:20, 5 May 2009 (UTC)


 * I DON"T GET IT
 * in stupid people language please ~bsb —Preceding unsigned comment added by 70.101.34.137 (talk) 20:14, 9 June 2009 (UTC)