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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 $P_0$.
…
- 22.1k
0
votes
Decide a manifold via its boundary
I'm wondering whether there isn't a much easier argument for 2-dimensional case, without using Seifert-van Kampen or the classification of surfaces (which is a middling hard theorem).
Fix the boundar …
- 22.1k
2
votes
Quandle colorings under Reidemeister moves
No.
For example, form a small loop with an R1 move on a strand coloured $x$, imagine the whole rest of the knot inside a small ball, and pass that ball on a loop-the-loop through the R1 loop. Then pe …
- 22.1k
6
votes
Generalised linking numbers (where they shouldn't be)
You'd think the definition of linking number would be common knowledge, wouldn't you? Actually, it's surprising how few places treat the topic carefully. The best references I know are:
H. Schubert …
- 22.1k
3
votes
Can you do surgery on framed tangles?
You can cut out a collection of 3-balls (a regular neighbourhood of the tangle), and glue them back in a different way. Such modifications are a part of the Montesinos trick. Montesinos uses them to p …
- 22.1k
8
votes
Accepted
Rational homology spheres and knots
For Question 1, I believe that the answer follows from:
Montesinos, José M. Surgery on links and double branched covers of $S^3$. Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. …
- 22.1k
4
votes
Compelling evidence that two basepoints are better than one
The most convincing example I have found of "two basepoints being better than one" is the incorrect statement of the main result of the following paper:
Garoufalidis, Stavros, and Andrew Kricker. "A …
- 22.1k
1
vote
Difference between Alexander polynomial and Blanchfield pairing
Clearly the signature, and more generally the Tristram-Levine signatures, would be needed.
- 22.1k
6
votes
Surgery diagram for the Seifert-Weber space
As pointed out by Ian Agol in the comments, the Seifert-Weber space is the 5-fold cyclic branched cover of the Whitehead link complement. You can therefore:
Untie the Whitehead link using $\pm 1$ fr …
- 22.1k
8
votes
2
answers
1k
views
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 …
- 22.1k
8
votes
2
answers
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 …
- 22.1k
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 …
- 22.1k
23
votes
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
This is a nice question!
Knot theory is in fact knot-complement theory, and a knot complement in S3 is a compact 3-manifold, while a knot complement in R3 is an open 3-manifold. Compact (or closed) 3- …
- 22.1k
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 …
- 22.1k
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 …
- 22.1k