Talk:Link (knot theory)

Too Specific

First I'd like to say that it is an attractive article with some nice pictures. However, it should be mentioned that there are other uses of the word link (and disambiguation). For example, consider a smooth function germ f with an isolated singularity at the origin:

$f:(\mathbb {C} ^{n},0)\to (\mathbb {C} ,0)$

Then we get a locally trivial fibration over the complex plane minus the origin. We consider

$F_{\epsilon }:=\{z\in \mathbb {C} ^{n}:f(z)=\epsilon ,\ \epsilon \neq 0\}\ .$

These are called the Milnor fibres. Then for small ε we intersect the milnor fibre with a sufficiently small hypersphere. The resulting intersection is called the link of the singularity. The topology of this link is of great importance. Moreover, varying ε along a closed path which does not pass through the origin gives a mapping from the homology of the link to itself, and induces what is called the monodromy. Dharma6662000 (talk) 20:16, 13 August 2008 (UTC)

Your example fits the basic definition of this article -- in fact the usage of "link" in singularity theory is derived from the manifold-theoretic usage of the word. I suppose "link" in the singularity-theoretic sense does not have to be a knotted collection of spheres but in general just a knotted collection of manifolds. This agrees with the historical usage of the word link. Is that what you're getting at? The knot theory articles on wikipedia all have that slant. This "link" definition is still a stub and biased towards plain-jane traditional 3-dimensional knot/link theory. Not much effort has been put into it yet. You're welcome to improve it. Rybu (talk) 21:37, 13 August 2008 (UTC)

You're quite right. Although I am feeling quite puzzled. I don't remember seeing the "more generally" section the last time I looked. There must have been a problem with my web browser, or my eyes. Sorry Dharma6662000 (talk) 21:51, 13 August 2008 (UTC)

I've touched it up a little. What do you think? Still needs some work. Rybu (talk) 21:59, 13 August 2008 (UTC)

It's getting better. I guess I'm just biased: I would love to see some mention of links in applications, e.g. singularity theory. For example a complex curve in the plane is a real two-dimensional body in four-dimensional space. The real link of a singularity on the curve is the intersection of a sufficiently small real 3-sphere centred at the singularity. If the singular point is isolated, then the real link is a one-dimensional manifold in a 3-dimensional space. Thus the link is of interest to knot theorists. And as you increase the dimensions of things you get these more general links. Some mention of braid monodromy would be nice too. I don't want to touch the article: it's very beautiful and my Wikipedia editing skills are next to nothing. If you need advice on the singularity theory then I can help you with that :o) Dharma6662000 (talk) 22:14, 13 August 2008 (UTC)

p.s. The milnor fibre is the wedged product of μ spheres where μ is the Milnor number of the singular function germ, i.e. the local multiplicity, i.e. the dimension of the local algebra. To compute this number you take the ring of function germs and quotient out by the Jacobian ideal, i.e. the ideal generated by all first order partial derivatives. Since the function is assumed to have an isolated singularity it follows that this is a finite dimensional vector space: the local algebra. For an example, consider $f(x,y)=x^{3}+y^{3}$. Then the Jacobian ideal is just

$J_{f}=\langle x^{2},y^{2}\rangle \ .$

The local aglebra is then given by

${\mathcal {A}}_{f}={\frac {{\mathcal {O}}(x,y)}{\langle x^{2},y^{2}\rangle }}=\langle 1,x,y,xy\rangle \ .$

It follows then that $\mu (f)=4$, and so the link is homeomorphic to the wedged product of four spheres. Dharma6662000 (talk) 22:31, 13 August 2008 (UTC)

IMO most of what you're talking about would be more appropriately placed in a topic like "fibred link". Links in general are not fibred so there's no monodromy to speak of -- there is a vague analogue in that you could talk about the action of the fundamental group of the link on the commutator subgroup or equivalently on the homotopy-fibre of the "abelianization map" from the link complement to the appropriate product of circles. But that's kind of a different topic. Rybu (talk) 01:09, 14 August 2008 (UTC)

Chainmaille

While not as scientifically "sexy" a subject as singularities, has there been any attempt to apply knot theory to chainmaille? — Preceding unsigned comment added by 67.142.178.20 (talk) 17:23, 3 January 2012 (UTC)