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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).
31
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4
answers
2k
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Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are …
- 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
6
votes
1
answer
139
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What is the original reference for disorientations on tangle diagrams?
There are several invariants whose "natural" domain is a category of disoriented tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the o …
- 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
18
votes
0
answers
492
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What do tangles teach us about braids?
A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$ …
- 22.2k
15
votes
1
answer
1k
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Why are Witten-Reshetikhin-Turaev invariants expected to be integral?
A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ …
- 22.2k
3
votes
1
answer
174
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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 …
- 22.2k
9
votes
1
answer
285
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Does the shortest path between two braids pass through string links?
One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.
This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, …
- 22.2k
3
votes
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
1
answer
413
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Is there a combinatorial version of PL ambient isotopy in dimension $>3$?
The classical PL Reidemeister Theorem reads:
Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by Reide …
- 22.2k
17
votes
3
answers
1k
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What is the state of the art for algorithmic knot simplification?
Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, a …
- 22.2k
4
votes
0
answers
234
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Is a generic link diagram semi-adequate?
Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution.
Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a li …
- 22.2k
10
votes
1
answer
1k
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Diagrammatic proof of unique prime decomposition of knots
Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident edg …
- 22.2k
10
votes
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
9
votes
1
answer
514
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Geodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?
I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.
We have …
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