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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).
3
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2
answers
333
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Is there a relative Pachner theorem?
Consider an $n$-dimensional PL manifold $P$ together with a closed subpolyhedron $P_0$ such that there exists a finite combinatorial triangulation of $P$ that restricts to a triangulation of $P_0$.
…
- 22.2k
8
votes
2
answers
1k
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Difference between Alexander polynomial and Blanchfield pairing
For a Seifert matrix $V$ of a knot $K$, the Alexander module has presentation matrix $V-tV^T$. The determinant of this matrix is the Alexander polynomial, which is the order of the Alexander module. I …
- 22.2k
8
votes
2
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858
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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 …
- 22.2k
25
votes
3
answers
2k
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What are the implications of the simple loop conjecture?
Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol.
Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow M …
- 22.2k
9
votes
2
answers
669
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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 …
- 22.2k
9
votes
2
answers
641
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Is more alternating always better?
While thinking about this interesting question asked by Dylan Thurston, it occured to me that in every case that I know, the closer a knot diagram is to being alternating, the better its properties. F …
- 22.2k
35
votes
4
answers
4k
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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 …
- 22.2k
82
votes
12
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15k
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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 …
- 22.2k
29
votes
3
answers
4k
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What would the slice-ribbon conjecture imply?
What would the slice-ribbon conjecture imply for 4-dimensional topology?
I've heard people speak of the slice-ribbon conjecture as an approach to the 4-dimensional smooth Poincare conjecture, and to t …
- 22.2k
5
votes
2
answers
371
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What is a higher genus analogue of the Pontryagin product?
Given a compact oriented aspherical $3$--manifold $M$ with torus boundary $\partial M\simeq T^2$ (e.g. a knot complement), the condition that the images in $\pi_1 M$ of basis $x,y\in \pi_1 T^2$ under …
- 22.2k
8
votes
3
answers
4k
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Seifert surfaces of torus knots
Does anyone know a nice description of a Seifert surface of a torus knot? I can construct such surfaces in band projection, but what I get is ugly and unwieldy. Is there some elegant description for S …
- 22.2k
8
votes
2
answers
828
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What is the Alexander polynomial of a point?
According to the Baez-Dolan cobordism hypothesis, an extended TQFT is determined by its value on a single point. This value a fully dualizable object of a symmetric monoidal $n$ category (a fully dual …
- 22.2k
8
votes
1
answer
934
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Freedman's work on non-simply-connected 4-manifolds
In the late 1970's and in the 1980's, Michael Freedman showed a relationship between the topological surgery problem in 4-dimensions, the slice problem for links, and the classification of non-simply- …
- 22.2k
17
votes
2
answers
3k
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Elegant proof that mapping class groups are generated by Dehn twists?
One of the foundational results about mapping class groups of surfaces is that they are generated by Dehn twists. A mapping class is a connected component in a space of diffeomorphisms, so another way …
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10
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3
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558
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Is a knotted trivalent graph determined by its set of unzips?
A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph in $\mathbb{R}^3$, modulo ambient isotopy. I believe they were first considered (in the quantum topological context, at least …
- 22.2k