Talk:Knot (mathematics)

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Mathematics rating: Start Class Mid Priority  Field: Topology

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)