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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).
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
31
votes
4
answers
2k
views
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 …
- 22.1k
6
votes
Construction of invariants of 4-manifolds with the Kirby calculus
The Witten-Reshetikhin-Turaev approach to constructing quantum topological invariants of $3$-manifolds is to define them on framed links and to prove invariance under Kirby moves.
There is a paper of …
- 22.1k
25
votes
3
answers
2k
views
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
6
votes
1
answer
139
views
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 …
- 22.1k
9
votes
2
answers
641
views
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
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
18
votes
0
answers
492
views
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 $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$ …
- 22.1k
15
votes
1
answer
1k
views
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 $\xi$ …
- 22.1k
3
votes
undergraduate handle decomposition. Reference
It sounds like Matsumoto's An Introduction to Morse Theory might be precisely what you are looking for. It gets to the heart of the matter quickly, and explains the main ideas very well. I taught a re …
- 22.1k
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 …
- 22.1k
1
vote
Accepted
Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geome...
The explanation of a Murasugi Sum in Chapter 4.2 of Kawauchi's book "A survey of knot theory" might clear up your doubts. "Figure 1", which you mention, indeed depicts a Murasugi Sum over a 4-gon.
To …
- 22.1k
9
votes
1
answer
285
views
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 $I^3$, …
- 22.1k
3
votes
Handlebody decomposition of a 3-manifold adapted to a link
Here is a rough construction of such a decomposition. First, glue another copy of $M$ to $M$ along their joint boundaries to get rid of the boundary, and then reduce your question to $S^3$ by performi …
- 22.1k
3
votes
2
answers
333
views
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