Talk:Knot theory

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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. Jkasd 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. Jkasd 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)

References

Shouldn't the references use footnotes rather than being inline like they currently are? -- Eraserhead1 <talk> 16:22, 15 January 2010 (UTC)

Knot Equivalence

The definition given there appears to be wrong. Actually it is defining "isotopy" rather than "ambient isotopy" and it is well-known (see e.g. the first pages of Burde-Zieschang: "Knots") that isotopy does not imply ambient isotopy. (Besides the map would only have to be injective level-wise, and moreover, of course it can not be injective as a map on [0,1] but rather on S^1.)--Kamsa Hapnida (talk) 12:05, 8 June 2015 (UTC) I have replaced that by a correct definition now.--Kamsa Hapnida (talk) 12:14, 8 June 2015 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Knot theory/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

History section could use some more references, especially to historically important papers. -- Jitse Niesen (talk) 05:08, 26 April 2007 (UTC) Knot definition The definition says that a knot is an injective continuous function K:[0,1]->R^3 with K(0)=K(1), but really K isn't injective if K(0)=K(1). — Preceding unsigned comment added by 2601:199:301:1691:C4E4:4553:198C:1587 (talk) 20:11, 12 January 2017 (UTC) History has been spun off into its own article, since people found it a bit distracting and long. --C S (talk) 22:26, 15 March 2009 (UTC)

Last edited at 22:26, 15 March 2009 (UTC). Substituted at 21:20, 29 April 2016 (UTC)

Recent Advance

This is not(!) my area but I came across this which sounds like a big deal: https://www.maths.ox.ac.uk/node/38304
Relates to "unknot" equivalence determination in near polynomial time by M Lackenby. Billymac00 (talk) 13:29, 3 February 2021 (UTC)