Drorbn - User contributions [en]

Talk:06-1350

2007-06-15T00:14:07Z

89.41.145.128:

Guys,

I thought it might be nice to establish a lively discussion board on which we can discuss, develop and clarify any, or all, aspects of material pertaining to the class! I'll try to get things started with something that I had on my mind.

From lectures, we've opted to get our teeth stuck into things from the outset, with the point being that a thorough discussion of the elementary concepts would eat up a substancial amount of time, from which the only outcome would be a rigorous confirmation of what we all already understood knot, and equivalence of knots, to mean.

Being eager, I thought I'd jot down a couple of sentences (having just finished the post, I can say that this is was a gross underestimate of length) indicating what I think the pedantic issues at stake to be, and was hoping that perhaps we could iron out a couple of things I find to be confusing.

As I see it, the first, and main, point that might warrent a discussion, is on the correspondence between smooth and peicewise linear embeddings of the circle in three space.

[Recall from the first lecture that we would like a knot to be an equivalence class of embeddings, where the embeddings can be taken to be peicewise linear or smooth and the equivalence relation is determined by ambient isotopy; namely f and g are said to be ambient isotopic embeddings if and only if we have a smooth/peicewise linear (depending on whether we're talking about smooth or peicewise linear embeddings) map H:S^3x[0,1]->S^3 subject to H(_,t) being a smooth/peicewise linear homeomorphism with H(_,0) the identity and H(f(S^1),1)=g(S^1).]

The first question is: what is the correspondence between the two theories? Clearly any peicewise linear knot may be smoothed at the corners, and any smooth embedding has a linear approximation. However, if we try to talk about isotopies of such knots we're talking about something living in S^3x[0,1]xS^3. I think smooth manifolds, of which the graph of H is one, can be triangulated, though I don't have a good reference for this. Further, if the peicewise linear isotopies could be "smoothed" then we could conclude that thinking of knots as being peicewise linear or smooth is really the same for our purposes. Does anyone know about this or have any good references?

The second question (which is really dependant on the first question) I had in mind was on the connection between representations of knots in the form of quadravalent planar diagrams up to Reidmeister moves, and knots up to ambient isotopy.

Firstly, well defined parallel projections of knots exist

[infact there is a lovely visual argument that show that almost all parallel projections are "nice" in the sence that the inverse image of points of the diagram have at most two preimages and such points are isolated... equate the projections with points on the sphere and observe that two linear segments of the knot project onto each other only on arcs of the sphere, and further that projections having preimage sets of cardinality greater or equal to two are isolated or again lie on an arc. Thus the measure of such a set is zero.]

Now ambient isotopies of peicewise linear isotopies of knots can be broken down into moves where you can either replace one segement of a knot with the other two edges of a triangle containing the fixed segment as an edge as long as the spanning triangle doesn't intersect the knot anywhere else and vica versa (a picture would be very handy here, but I don't really know how to use any such feature yet), and consequently we see the Reidemister moves of planar diagrams all correspond to isotopies. The other direction is Reidemeister's famous theorem (the equivalence that we assumed axiomatically) and is quite a complicated argument. The question that arises in my mind is that if the smooth and peicewise linear theories are not equivalent [see question one], then it would seem that Reidemeister's proof may not carry over!

I'd love to have this put straight in my mind, and hope that someone can help me via a reference or authorative confirmation.

Sorry this was long winded, confused and above all practically irrelevant to the course!

Fionntan

My life's been basically dull these days.
I haven't gotten much done these days.
Today was a complete loss.

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07-1352/About This Class

2007-06-15T00:13:38Z

89.41.145.128: /* Crucial Information */

{{07-1352/Navigation}}

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===Abstract===
This is a continuation of [[06-1350]] and the abstract remains the same:

An "Algebraic Knot Theory" should consist of two ingredients
# A map taking knots to algebraic entities; such a map may be useful, say, to tell different knots apart.
# A collection of rules of the general nature of "if two knots are related in such and such a way, their corresponding algebraic entities are related in such and such a way". Such rules may allow us, say, to tell how far a knot is from the unknot or how far are two knots from each other.

(If you have seen homology as in algebraic topology, recall that its strength stems from it being a functor. Not merely it assigns groups to spaces, but further, if spaces are related by maps, the corresponding groups are related by a homomorphism. We seek the same, or similar, for knots.)

The first ingredient for an "Algebraic Knot Theory" exists in many ways and forms; these are the many types and theories of "knot invariants". There is very little of the second ingredient at present, though when properly generalized and interpreted, the so-called '''Kontsevich Integral''' seems to be it. But viewed from this angle, the Kontsevich Integral is remarkably poorly understood.

The purpose of this class will be to understand all of the above.

===Warning===
'''This class is not for everyone.''' An old rule says one should not give a class on one's own current research. Here we will break that rule with vengeance - the class won't just be about current research, it will be about research that had not been done yet. Our purpose will not be to paint a beautiful picture of an established field, rather, to learn about the parts that may one day fit into and create such a beautiful picture, or may not. The parts are pretty in themselves and will force us to tour a number of deep mathematical fields. But by the nature of things, the presentation may well be confused and frustrating. If that scares you, or if all you need is a sure credit, '''do not take this class.'''

The stress of giving a coherent description of a non-existent subject will be too much for me. To mask this, whenever I will need a break we will branch off into asides, some more relevant and some less. Possible topics include: categorification, more on Chern-Simons and Feynman diagrams, more on Stonehenge pairings and configuration spaces, the Århus integral, multiple <math>\zeta</math>-numbers and more.

===Prerequisites===
Having taken [[06-1350]] or having made a heroic effort to catch up.

===Wiki===
The class web site is a wiki, as in [http://www.wikipedia.org Wikipedia] - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, solution to open problems, whatever. Some rules, though -
* This wiki is a part of my ([[User:Drorbn|Dror's]]) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
* If there's no specific reason for your edit to be anonymous, please log in and don't have it anonymous.
* Criticism is fine, but no insults or foul language, please.
* I ({{Dror}}) will allow myself to exercise editorial control, when necessary.
* The titles of all pages related to this class should begin with "07-1352/", just like the title of this page.
Some further editing help is available at [[Help:Contents]].

===Good Deeds and The Final Grade===
The grading scheme will be announced a few weeks into the class. Whatever it will be, it will allow you to earn some "good deeds" points throughout the semester for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:
* Solving an open problem.
* Giving a class on one subject or another.
* Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
* Taking class notes in nice handwriting, scanning them and posting them here.
* Formatting somebody else's class notes, correcting them or expanding them in any way.
* Writing an essay on expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
* Doing anything on our [[07-1352/To do]] list.
* Any other service to the class as a whole.

'''Important.''' For your good deeds to count, you '''must''' do them under your own name. So you must set up an account for yourself on this wiki and you must use it whenever you edit something. I will periodically check [[Special:Recentchanges|Recent changes]] to assign good deeds credits.

===Homework===
There will be 0-3 problem sets. I encourage you to discuss the assignments with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions.

06-240

2007-06-14T22:58:03Z

89.41.145.128:

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