Talk:Knot theory

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This article is within the scope of WikiProject Mathematics.
Mathematics grading:	B+ Class	High Importance	Field: Geometry and topology

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

Still to come:

• Gauß diagrams[1]
• 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!

—Herbee 13:17, 2004 May 16 (UTC)

Contents 1 broken link 2 You/Me 3 List of knot theory topics 4 Adding knots 5 New To Advanced Math 6 Rewrite to cover the subject better? 7 citation for Haken's recognition algorithm 8 Possible mistake 9 knot theory rewrite

[edit] broken link

[2] 

[edit] 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, Jun 4, 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. --Chan-Ho (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). --KSmrq^T 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. --KSmrq^T 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)

[edit] 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 Jun 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)

[edit] 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. --Chan-Ho 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. --Chan-Ho 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?" : )

[edit] 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 26 Jan 2006

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)

[edit] 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. --Chan-Ho (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)

[edit] 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. --Chan-Ho (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). --Chan-Ho (Talk) 06:10, 11 October 2006 (UTC)

[edit] 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. --Chan-Ho (Talk) 19:34, 11 January 2007 (UTC)

[edit] knot theory rewrite

I started a big rewrite of the article at User:Chan-Ho_Suh/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. --Chan-Ho (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. --Chan-Ho (Talk) 07:26, 19 March 2007 (UTC)