Talk:Knot (mathematics)

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In mathematics, knotting in general refers to any situation where a manifold M embeds in another manifold N is more than one way (up to isotopy). So technically, a diffeomorphism of a manifold f : M --> M which is not isotopic to the identity could be considered a "knot". One of the most neglected knot theories is the theory of surfaces in the 3-sphere. It would be nice if this wiki could eventually have parts that reflect this. I made a few small modifications but they'll need polishing.

It's true that "knotting" is a more general notion, but "knot" meaning an embedded circle in S^3 is undoubtedly far more often used than other contexts. Even for knot theorists "knot" really could also mean say an embedded circle in some 3-manifold, but the S^3 case dominates the literature.

I think the material on "knot" beyond S^1 in S^3 should be properly made into either separate sections or articles. Almost everybody coming to this page will expect it to be on S^1 in S^3. The current mixing of concepts is rather confusing to read. --C S (Talk) 06:59, 22 March 2007 (UTC)

This wiki is titled "knot (mathematics)" so IMO it should give the mathematical definition of a knot. I agree knots in S^3 are the most commonly studied and accessible, but it doesn't make that subject of all of knot theory, it just means it's well-studied. I do think largely focusing on knots in the 3-sphere is a good idea, but the proper definition should be mentioned. Making another section in the current article is probably the best thing to do, but the intro will have to be adapted. I think I see your point -- you want the article to be accessible so that the reader doesn't have to wade through the definitions of manifold, embeddings, spheres, etc, before understanding the definition of a knot in S^3, because it is very intuitive. IMO perhaps a historical perspective, like an etymological definition of the word knot in mathematics might be the best approach. First knots in S^3, then the knotting phenomenon in general, and basics about knots in higher dimensions. Things like the Alexander/Schoenflies theorem (no co-dimension 1 knots in S^2) could be mentioned.

I've made a few changes to take into account your comments. By all means, more work needs to be done. I'll peck away at this for a few more days... I think maybe some pictures would be appropriate. A few links to Dror Bar-Natan's wiki might be appropriate and maybe a mention of what exactly geometrization means for knot complements.

Yes, the page is much improved. Thanks for the effort! Your ideas sound good; pictures would be nice, but are always scarce... --C S (Talk) 22:12, 22 March 2007 (UTC)

higher dimensions for dummies

In Talk:Fourth dimension, in a list of points that make 4space interesting, someone mentioned that 2-surfaces can be knotted in 4-space and gave the Klein bottle as an example, calling it a knotted 2-sphere. But a knot in the usual sense is made of a string which is topologically a 1-sphere, and the K-bottle is topologically not S2. So – Is the Klein bottle a knotted S2 or not? If not, what's the 4space analogue to the trefoil knot? —Tamfang (talk) 06:05, 1 May 2010 (UTC)

Could you be more specific in your reference to Talk:Fourth dimension ? The Klein bottle is certainly not a sphere. But there are knotted spheres in $\mathbb {R} ^{4}$. Standard ways to construct these embeddings are called "knot spinning". One direct analogue to the trefoil would be called the Artin spin of the trefoil (or the 0-twist spin of the trefoil). It's analogous to the trefoil in that its complement has fundamental group isomorphic to the fundamental group of the trefoil complement. Rybu (talk) 14:28, 14 May 2010 (UTC)

Sorry, Talk:Fourth dimension#Dimensional analogue. —Tamfang (talk) 17:34, 15 May 2010 (UTC)

The comments about Klein bottles being knotted spheres is a mistake. The rest of that line is okay. 2-manifolds do knot in 4-space. Rybu (talk) 17:12, 16 May 2010 (UTC)

Merger proposal

I have just stumbled upon the article about knots and graphs (which will be referred to as the source article), and realised that it may be suitable for merging with the knots (destination) article. Some of the reasons include:

• Most of the content in the source article is not covered in the destination article. Examples of such content include the medial graph of a knot and the Reidemeister moves.
  • Much of it is redundant for Wikipedia itself (specifically, drawing nice knots).
• I foresee that the resulting article would be of a comfortable size for a reader (destination article is too small and source has too much unnecessary content that, when removed, makes it too small as well).

FREYWA 07:36, 15 October 2011 (UTC)

Merge with knots and graphs

I've merged the articles Knot (mathematics) and knots and graphs. Most of the graphs stuff is in a separate section, but I intertwined some of the definitions (like projections and Reidermeiester moves). Also, standardised the terminology, as R³ and E³ were being used interchangeably (with no explanation that they were the same), and S³ and S₃ were being used interchangeably. Joseph2302 (talk) 16:59, 18 March 2015 (UTC)