Timeline for What is known about links with a countably-infinite number of tame components?
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
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| Feb 1, 2011 at 23:11 | vote | accept | Scott McKuen | ||
| Feb 1, 2011 at 21:54 | comment | added | Scott McKuen | Sergey - thanks, that would be great. My contact address is isotropy-at-gmail. | |
| Feb 1, 2011 at 17:03 | comment | added | Sergey Melikhov | (cont'd) Scott, if you are interested to discuss this approach in more detail, I'd be glad to continue by email. As for (a), I think the real question is to get a more combinatorial understanding of smooth topology. Robert MacPherson has had a program on "combinatorial differential manifolds" published in a conference proceedings in mid-90s. Unfortunately the Ann. Math. paper of Biss claiming advances on this program is in error, as explained by N. Mnev in arXiv:0709.1291. I do have something of an alternative program going in the direction of polynomial compatification of configuration spaces | |
| Feb 1, 2011 at 16:45 | comment | added | Sergey Melikhov | In fact, model theory has a rather good understanding of PL topology: see van den Dries' book "Tame topology and o-minimal structures", A. J. Wilkie's paper (in English) in Seminaire Bourbaki (no. 985, 2007) and M. Shiota's recent papers on the arxiv, including arXiv:1002.1508. I'm not aware of anything about model theory of a) smooth topology or b) TOP (i.e. continuous) topology of manifolds. But the similarity that we are seeing between Fraisse limit the back-and-forth arguments, on the one hand, and pro-categories and shape theory on the other hand, seem to be pointing at an approach to (b) | |
| Feb 1, 2011 at 16:23 | comment | added | Sergey Melikhov | Scott, 1) I guess every component would have infinitely many other components converging to it. 2) Reidemeister moves axiomatize PL (piecewise-linear) ambient isotopy of finite PL links, and also of proper infinite links. Proper links have no components converging to other components. This is one key technical point. Another is that although topological ambient isotopy of PL links implies PL ambient isotopy, it is a rather difficult theorem. It was proved by Bing and Moise in the 50s and is exposed in their monographs from 70s/80s, and AFAIK, not in any other textbook. | |
| Feb 1, 2011 at 4:54 | comment | added | Scott McKuen | Actually, to the extent ambient isotopy is first-order axiomatizable in a logical language for knots where you can also do Fraïssé limits, the limit should have the first-order properties, I think. This problem of losing actual ambient isotopy would seem to imply that you can't define ambient isotopy in a first-order way for structures on links. That's perplexing to me, because the Reidemeister moves smell a lot like first-order conditions, and my understanding was that they axiomatized knot equivalence in just this way. I must be missing a key technical point here. | |
| Feb 1, 2011 at 4:48 | comment | added | Scott McKuen | @Bill - ugh. The obstruction I was worried about was finding a first-order language that made the maps of sublinks into links become actual logical embeddings. The possibility that in passing to the limit you lost ambient isotopy hadn't occurred to me. Should this only happen in the case where the limit ended up with a component that had infinitely many other components linked directly to it, or is it worse than that? | |
| Jan 31, 2011 at 5:23 | comment | added | Bill Thurston | @Scott: Thanks for adding the model theory background. I'm likely to be confused, but it's not obvious to me whether the usual theory of links fits readily into this framework, since topology is conventionally not set up as a first-order theory. In particular, I'm worried about giving a topological meaning to the limiting universal object of the increasing union of links. I can imagine a limiting mess of a perhaps dense union of embedded circles, but the structure of the whole, as distinguished from all finite sets of link components, wouldn't appear to have any naturality (without more). | |
| Jan 31, 2011 at 3:11 | comment | added | Scott McKuen | Ryan, thanks for this example - Joel's answer is right about what I'm trying to get at. I've added some context above that spells out the difference between the intended structure and a disjoint union of everything. | |
| Jan 31, 2011 at 3:09 | history | edited | Scott McKuen | CC BY-SA 2.5 |
adding details on the intended construction
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| Jan 31, 2011 at 2:51 | comment | added | Joel David Hamkins | I'm not sure exactly what Scott wants, but the model-theoretic constructions simply add objects so as to witness all possible relations with what has been constructed so far. To build the rational order from finite orders, you systematically add points between all your current points, and to build the random graph, you add new vertices in every finite relation to the current vertices. So I imagine that he wants somehow to systematically add loops that intertwine with the current loops in every conceivable fashion, realizing as many finite configurations along the way as possible. | |
| Jan 31, 2011 at 2:33 | comment | added | Sergey Melikhov | Ryan, maybe they want an increasing sequence of finite links $L_1\subset L_2\subset\dots$ such that whenever some $L_n$ occurs as a sublink of a finite link $L$, then this $L$ must be a sublink of some $L_m$ relative to $L_n$ (that is, the identity on $L_n$ must extend to a self-homeomorphism of $\Bbb R^3$ sending $L$ onto a sublink of $L_m$). If so, then the union of all the $L_n$ is definitely not going to be a proper link (in the sense of my answer below). | |
| Jan 31, 2011 at 2:12 | answer | added | Sergey Melikhov | timeline score: 2 | |
| Jan 31, 2011 at 2:09 | answer | added | Bill Thurston | timeline score: 3 | |
| Jan 31, 2011 at 1:42 | comment | added | Ryan Budney | What does that mean? | |
| Jan 31, 2011 at 1:28 | comment | added | Joel David Hamkins | Rather than merely having separate copies, I think he wants gradually to add new links that relate to the current links in all possible ways. This is the essence of the model-theoretic constructions... | |
| Jan 31, 2011 at 0:48 | comment | added | Ryan Budney | There are only countably many link types (provided you're dealing with links with finitely many components), so you can simply list them all and put them in disjoint balls in $\mathbb R^3$. It's not clear to me what your comments about model theory and Reidemeister moves are getting at. | |
| Jan 31, 2011 at 0:17 | history | asked | Scott McKuen | CC BY-SA 2.5 |