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Jul
20 |
asked | Complexity of mappings (forms) in R. Thom's “Structural stability and morphogenesis” |
May
24 |
awarded | Critic |
Apr
25 |
accepted | What does it mean that homotopy is generic? |
Apr
25 |
asked | What does it mean that homotopy is generic? |
Jul
3 |
awarded | Commentator |
Jul
3 |
comment |
Fundamental problems whose solution seems completely out of reach
The only one invariant of knots which seems to be complete and many people beleve in it is Vasilliev' invariants, discovered by russian mathematician Vasilliev in 90-th. It was based on the ideas of singularity theory. M. Kontsevich also took a hand on them, constructed Kontsevich integral, and the whole question "redused" to combinatorics. But... My idea is that knot "nature" is still not known, I mean, it seems that it should inspire us to construct new mathematical tool. In this sence, due to Poincare, it is a good question. Sorry for my English. It is not my native language. |
Jul
2 |
comment |
Fundamental problems whose solution seems completely out of reach
That's right. But this algorithm depends on knot crossings so is hard to compute. And it doesn't close the classification problem. Given an arbitrary knot, you cannot, in general,determine its isotopy class neither by this algorithm nor by other invariants. P.S. As I know, Haken didn't finish the work on this algorithm. It was done by russian mathematician S.V.Matveev. |
Jul
2 |
comment |
Diffeomorphisms of a surface in terms of generators.
Sorry for my terminology. I didn't take care of it hoping that if someone is familiar with this then he could help, without digging in details. |
Jul
2 |
comment |
Diffeomorphisms of a surface in terms of generators.
Thank you for your response. I will comment your answer and upper questions here. I am considering a particular diffeomorphism (up to isotopy to identity) of a surface F given by a function f: F --> F. Can I write down a presentation of this f in terms of Dehn twists? For example, given an involution x |--> -x,can this be made? |
Jul
1 |
comment |
Diffeomorphisms of a surface in terms of generators.
Of course I mean generators of homeotopy group. |
Jul
1 |
asked | Diffeomorphisms of a surface in terms of generators. |
Jul
1 |
awarded | Teacher |
Jul
1 |
comment |
Fundamental problems whose solution seems completely out of reach
I mean a classification of knots under the isotopy. Just formulated it in physical setting to show its simple nature. A physical type problem is also unsolved. There are some development here, uncluding algorithms, depending on the thickness of a knot. |
Jul
1 |
answered | Fundamental problems whose solution seems completely out of reach |
Jun
29 |
awarded | Scholar |
Jun
29 |
comment |
In which cases a fiber bundle over a circle is a graph-manifold?
Thanks a lot!!! |
Jun
29 |
accepted | In which cases a fiber bundle over a circle is a graph-manifold? |
Jun
29 |
asked | In which cases a fiber bundle over a circle is a graph-manifold? |
Jun
24 |
comment |
Minimal piecewise-linear knot diagram
Yes, it seems that the notion of a "stick number" is what I'm looking for. Yes, by "can we find" I mean an algorithm. It is obvious that this stick number is an invariant, so it seems interesting to look for an invariant, depending on this number, or think about knots with the same stick number. Thank you all for answer. It is sometimes difficult to find something, because of terminology. |
Jun
23 |
asked | Minimal piecewise-linear knot diagram |