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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
votes
1
answer
236
views
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
6
votes
Tiling of genus 2 surface by 8 pentagons
Ian answered the second question as asked, but in case you meant to ask a different question: there is not always a symmetric tiling by regular polygons of the given type, even if those restrictions h …
- 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
views
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
views
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
views
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
6
votes
Accepted
Thickening graphs to get honest actions
Every automorphism of $F_n$ can be realized by an automorphism of the 3-dimensional handlebody of genus $n$, obtained by attaching $n$ 1-handles to a 3-ball.
- 10.1k
5
votes
Is there a version of Seiberg-Witten-Floer or Heegard-Floer homology for 3-manifolds with bo...
Andy Manion has already plugged our answer for Heegaard Floer homology. On the Seiberg-Witten side, not as much is known, but Tim Nguyen's thesis starts to attack the problem. However, there isn't a …
- 10.1k
27
votes
Accepted
Usefulness of using TQFTs
All the answers so far have focused on 3 dimensions, but the answer is much more striking in 4 dimensions. Freedman's theorem tells you that classical homology invariants give you complete informatio …
- 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
49
votes
Elegant proof that any closed, oriented 3-manifold is the boundary of some oriented 4-manifold?
I know of several different arguments. You can decide which one you think is most elegant...
Rohlin's argument, which is actually quite geometric. You start with an immersion of the 3-manifold in $ …
- 10.1k
8
votes
1
answer
392
views
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
views
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
6
votes
Do the results of (1/n)-surgery determine the link?...
If the orginal link $U_1 \cup U_2$ was hyperbolic, the answer is yes. For large enough $n$, $S^3 \setminus K(n)$ will also be hyperbolic, and will approach $S^3 \setminus (U_1 \cup U_2)$ in the Grom …
- 10.1k
7
votes
1
answer
310
views
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