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Results tagged with gt.geometric-topology
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user 2051
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
24
votes
3
answers
3k
views
Elevator pitch for the Virtual Fibering Theorem
There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm guess …
- 4,713
3
votes
1
answer
174
views
Tait conjectures for alternating w-links
The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state:
Any reduced diagram of an alternating link has the fewest possible crossings.
Any two reduced …
CommunityBot
- 1
14
votes
Accepted
Knot diagrams, sets of moves and equivalence relations
Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves.
The study of claspers. For example, $C_k$-moves are a sp …
- 1,446
30
votes
2
answers
3k
views
The difference between a handle decomposition and a CW decomposition
Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it induc …
- 2,821
13
votes
Is the HOMFLY Polynomial the best knot invariant?
What about the coloured HOMFLYPT? It's clearly stronger than the HOMFLYPT. Whether it is a complete knot invariant is (I believe) open. Mutation preserves the HOMFLYPT polynomial. The 2-variable HOMFL …
- 2,821
2
votes
Is Murasugi's conjecture still open?
See also this paper by Jong which reproves the Ozsvath-Szabo result combinatorially, using Stoimenow's generators for knots of canonical genus 2.
The interesting question which is lurking in the backg …
- 2,821
5
votes
Kirby calculus and local moves
I think that the answer to your question is yes EDIT: if you allow local to mean "local within a thickened surface" or allow local moves between tangles whose strands may not be part of the surgery li …
- 2,821
35
votes
4
answers
4k
views
Extending a diffeomorphism of the sphere $S^2$ to the ball $D^3$
A fundamental result in three-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of the two-sphere $S^2$ ex …
- 28.2k
21
votes
1
answer
2k
views
How are the Conway polynomial and the Alexander polynomial different?
Background story:
I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he des …
- 4,713
82
votes
12
answers
15k
views
Compelling evidence that two basepoints are better than one
This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his …
- 35.5k
8
votes
Kirby calculus and local moves
If you didn't like my first answer, here's a different one; again, slightly changing the question. This answer is better suited to the way the Kirby theorem is used in quantum topology.
Consider the s …
- 4,713
8
votes
2
answers
858
views
Are there any very hard unlinks?
This question is closely related to a question of Gowers: Are there any very hard unknots? .
I'm thinking about how to create interesting knots from small numbers of local moves on unlinks. The "stand …
- 35.5k
23
votes
Accepted
Proofs of Kirby's theorem
There is Bob Craggs' 1974 proof, which was never published. It relies on Wall's result, that any two 2-handle cobordisms between S^3 and itself are stably homeomorphic if the associated bilinear form …
- 35.5k
9
votes
2
answers
669
views
Does every knot contain all four vertices of an isosceles trapezoid?
I ask this question with some trepidation, because it may be trivial and/or of entirely recreational interest.
Erika Pannwitz proved in 1933 that every non-trivial knot contains a quadrisecant (four p …
- 35.5k
8
votes
What are some of the big open problems in 3-manifold theory?
The Virtually Fibred Conjecture, and related problems.
For a weaker definition of 3-manifold topology, I think the Andrews-Curtis conjecture is a key problem. Also, anything which relates to the class …
- 4,713