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Results tagged with gt.geometric-topology
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user 5010
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).
6
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1
answer
236
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Reference request: Can iterated torus links be mutated?
I believe that most iterated torus links cannot be changed non-trivially by a Conway mutation, as follows. If you look at the JSJ decomposition of the double-branched cover, then each satellite torus …
- 10.1k
18
votes
3
answers
630
views
Classification of tangles?
Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't rationa …
- 10.1k
49
votes
2
answers
4k
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Can knot diagrams be monotonically simplified using under moves?
It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger …
- 10.1k
10
votes
2
answers
238
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What are the possible linking matrices of a quasi-positive link?
I was surprised recently to come across a 3-component link where the linking number of two of the components was negative. For a while I thought I had made a mistake, then I thought a little more and …
- 10.1k
5
votes
1
answer
271
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Is every triangulation of a Euclidean ball by convex tetrahedra shellable?
Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable?
I know that if $ …
- 10.1k
12
votes
1
answer
713
views
Injectivity of the Dehn-Nielsen-Baer map?
If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class …
- 10.1k
8
votes
1
answer
392
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To find a point in Teichmüller space or measured foliation, how many lengths of curves do yo...
To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in …
- 10.1k
10
votes
2
answers
579
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Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bo...
The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$,
$$\mathrm{sl}(K) \le - \chi(\Sigma)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-l …
- 10.1k
7
votes
1
answer
310
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Can a surface group act on a finite-valence simplicial tree?
Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree.
Thi …
- 10.1k
13
votes
2
answers
482
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Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the s …
- 10.1k
14
votes
1
answer
1k
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Who proved that two homotopic embeddings of one surface in another are isotopic?
If $\Sigma_1$ and $\Sigma_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma_1 \hookrightarrow \Sigma_2$ are two orientation-preserving embeddings that are homotopic, then …
- 10.1k
54
votes
5
answers
2k
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Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge le …
- 10.1k