Talk:History of knot theory

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 Field:  History of mathematics

A new beginning[edit]

See Talk:Knot_theory#Featured_article_candidate. We might as well start by including the history section from knot theory, but I think it is really beneficial to start completely from scratch. Moritz Epple, a historian, has written about the early history of knot theory fairly comprehensively, so there are better sources now than when the history section was written (I basically used Dan Silver's article). --C S (talk) 05:53, 26 May 2008 (UTC)

Some specifics:

  • M. Epple, Topology, Matter and Space I: Topological notions in 19th-century natural philosophy. This paper is very interesting because it suggests that it was the study of Helmholtz's work and Thomson's vortex atom theory that brought Listing's Topologie to wider attention in England and thus disseminated topology to a much larger audience. I can't find a free copy online (although somehow I did it before, email me for copy).
  • M. Epple, Branch Points of Algebraic Functions and the Beginnings of Modern Knot Theory [1] From the abstract, it seems to be about the origins of computations of the knot group (starting with Wirtinger), which developed out of studies of algebraic functions. This looks like an important part of the history that is left out in the current history subsection.
  • There seem to be a variety of history articles on Thomson's vortex atoms. Perhaps it's time to create a separate article on that too.

I don't want to go overboard with this, but I suggest including some of the highlights of the Epple papers I listed as they place knot theory in a historically richer background. --C S (talk) 21:19, 26 May 2008 (UTC)

I was drafting this article here, but I agree that it is nice to start fresh, so I've just added the new parts I'd already written. Jkasd 05:13, 27 May 2008 (UTC)

Haken's algorithm too complicated?[edit]

The reports of its complications, like the one in Adams' The Knot Book are mistaken. The algorithm can be easily described in one paragraph. Triangulate the knot complement. Generate the matching equations. These are integer linear equations. Now search for a minimal Hilbert basis, which are a finite set of nonzero integral solutions to the equations. Do an "Euler characteristic" calculation for each solution vector (this is simple). If one of them has Euler characteristic one, then stop and return "unknotted". If none of them do, return "knotted". (Explaining the details of this and why it works would only take several pages)

Certainly nothing is stopping anyone from doing this on a computer and in fact it has been done by various people. Computer programs exist that implement various parts of this. For example, there are some programs that create triangulations from knot diagrams. This can then be fed to something like Regina) which can then find a finite set of solutions which must contain an unknotting disc if it exists. Assertions of the algorithm's complicatedness can be more fruitfully interpreted to be about the complexity of the algorithm. But actually, that is unknown and an active area of research. The algorithm is good enough to show unknotting is in NP, but the worst case bounds are exponential. Of course, that doesn't really tell you what the average performance may be. There are also various ways to try and speed up the different parts. One thing that is much faster than Haken's original approach is to enumerate only the vertex solutions, which is exactly what Regina does (there's a proof that an unknotting disc, if it exists, is represented by such a vertex solution).

Another thing...I'm not sure what this is doing in the "modern resurgence" section anyway. The resurgence described in the original history section was referring to the stuff that happened in the period after Thurston and Jones work. I think Haken's work would be considered to be in a prior 'classical' period. --C S (talk) 07:23, 29 May 2008 (UTC)

Sorry, I guess I should have checked with other sources first. Thanks for fixing it the suggestions. Jkasd 17:02, 30 May 2008 (UTC)
So is Adams' book generally a good source? I'm sure all books have mistakes, but is it reliable enough to use most of the time? Jkasd 17:12, 30 May 2008 (UTC)
Don't worry about it. Adams is a good source in general. There shouldn't be a need to thoroughly vet passages from Adams with other sources. This error, unfortunately prominently displayed, was fairly natural to make at the time (1994). The newest edition has a page of error corrections, but I don't have it so I don't know if it includes this. There are probably some little minor errors here and there still. One I noted that might be hard to catch is Figure 5.29 (caption: two knots with the same volume): as can be verified by using SnapPea, they don't! (the one on the right is supposed to be the (-2, 3, 7) pretzel knot. --C S (talk) 23:31, 30 May 2008 (UTC)


Progress on this article has been slow, and the focus should be on the possible FA candidacy of knot theory anyway. One possibility is to just forget all this sectioning, creating a lede, etc. and just copy over the entire history section from knot theory to this article. Wikipedia is never finished, and I'm sure people will work on this article later. --C S (talk) 00:21, 4 June 2008 (UTC)

Well, it might be too early to implement the change, but I went and did it anyway. Feel free to revert in whole or part. Improvements are necessary but the question is whether people are really willing to redo the article from scratch. I think not (correct me if I'm mistaken!). Note I did incorporate a few things that have been added, such as the topological quantum computation. --C S (talk) 22:45, 4 June 2008 (UTC)

Reliable sources[edit]

This has lots of interesting details about the early history of knot theory, but I'm not sure how reliable the source is. Actually, even though I've read WP:RS and the likes, I'm not sure when a source is considered reliable, some general guidelines would be appreciated. Jkasd 18:13, 9 June 2008 (UTC)

The general principle is that the greater the level of fact-checking and scrutiny, the more reliable the source is. I think you probably already know how to determine the reliability of the source, given that WP:RS emphasizes published, peer-reviewed sources. The linked paper is an undergraduate honors thesis written for a math degree. From that we can reasonably infer that probably one person besides the author checked it (his major advisor). We can also infer that the level of scrutiny is probably far less than that for say, a Ph.D. thesis, which would have an entire committee of people who would consider it a blemish on their reputation if they missed a big mistake. Also, given that the purpose of undergraduate theses like this is to have the undergraduate learn mathematics, I'm not sure if his advisor would even have really bothered to thoroughly scrutinize the historical overview section. This is similar to how even a good math paper in a prestigious journal may not check the historical remarks as throughly as the mathematics.
Of course, you probably already know all this, so you may just be asking what is "generally accepted". Ok, well, from what I've noticed, an undergraduate thesis would never pass (if you included this source in an FAC it would never pass). A Ph.D. thesis would usually be ok, but a mathematics one would probably not be considered reliable for another subject like history (even if it includes a bit of history), and so forth. Real journal articles written by well-known historians are obviously the best for historical topics, but a lot of times you can source things to technical papers that have some history. RS checkers generally like books by well known publishers, so books are always a good bet.
Unpublished e-prints are not generally considered reliable, but if you argue that the author is well-respected in the field, you can probably get others to buy it, especially if the e-print is a survey or expository article (versus claiming to prove some big unsolved problem).
By the way, you really need to format the refs you include. I've noticed you only tend to give source info for sources not on the Internet, but you need to write the source info for stuff you found through the Internet also; just giving the link is considered bad form. Also, from what I read, the linked thesis took everything from the Epple and Hoste works I've mentioned above. He also references the Math Intelligencer article by Hoste, Thistlethwaite, and Weeks; I expect it's very nice and readable (MI articles usually are), so if you can get a copy of that, I think you'll find it very helpful.--C S (talk) 21:29, 9 June 2008 (UTC)
I know my referencing has been bad, which is why I asked first instead of just sticking it in which is what I would have usually done. I'm not sure what you mean by formatting, let's take the ref I put in on this page <ref></ref> which is just the scientific american article. How should I format it with the source info and stuff? Anyways, thanks. Jkasd 22:18, 9 June 2008 (UTC)
Well, I wouldn't say your referencing has been "bad"! About the formatting issue, take a look at User:Jkasd/Knot_theory_draft. Is there a reason you included the author, title, and publisher info for Kauffman's book and MathWorld but not the others? For the Stoimenow reference, I would include something like: A. Stoimenow, "Tait's conjectures and odd amphicheiral knots", 2007, arXiv: 0704.1941v1. It doesn't seem to have been published, but often arXiv papers will also have been published somewhere so you can dig up that info and include it usually (but be careful that the published version isn't really different if you do this, otherwise it could lead to big confusion). Also, for arXiv references, I prefer to link to the abstract page (like I did in my example) because arXiv offers several different formats to download the paper.
For the Scientific American article, I would follow a standard format like:
Collins, Graham (April 2006). "Computing with Quantum Knots". Scientific American. pp. 56–63.  Check date values in: |date= (help).
the knot theory article uses Harvard referencing but Poincaré_conjecture uses extensive footnoting and reference templates. the latter article has a mix of different kinds of references, so that should give you an idea. --C S (talk) 23:20, 9 June 2008 (UTC)
I found the Math Intelligencer article online. It looks very nice. You will need some kind of academic/university access to get to it, but I'm guessing you won't have a problem with that... --C S (talk) 23:24, 9 June 2008 (UTC)
I guess I do tend to be kind of hard on myself. Thanks for the help. I couldn't find the article you mentioned. Jkasd 02:21, 10 June 2008 (UTC)