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Results tagged with gt.geometric-topology
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user 11142
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).
5
votes
3
answers
522
views
Triangulations of 3-manifolds in Regina and SnapPy
I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a …
- 96.4k
7
votes
1
answer
355
views
Number of (distinct) knots with a bounded number of crossings
The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternati …
- 96.4k
6
votes
1
answer
300
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Generating prime knots (in order)
In this really cool paper https://arxiv.org/abs/1612.03368, A. Malyutin shows that the probability that a random prime knot of up to $N$ crossings (as $N$ goes to infinity) is not generically hyperbol …
- 96.4k
13
votes
0
answers
370
views
Euler characteristic of *hyperbolic* orbifolds
This is a follow-up to this question. Which rational numbers arise as Euler characteristics of orbifold quotients of $\mathbb{H}^n?$ The answer is not even clear for $n=2.$ It is clear that the Euler …
- 96.4k
1
vote
0
answers
86
views
Dirichlet domain complexity
Suppose I have a subgroup of $SL(2, \mathbb{Z})$ given by explicit matrix generators, and my goal in life is to construct a Dirichlet domain (for, say, everyone's favorite basepoint $\sqrt{-1}$). Is t …
- 96.4k
5
votes
0
answers
129
views
Images of the $3$-dimensional solvable geometry
This is more of a mathematical art question. Most people are familiar with the mathematical movie "Not Knot", which explores a hyperbolic $3$-manifold. My question is: are there any images along the s …
- 96.4k
12
votes
1
answer
416
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Which compact 3-manifolds with boundary embed in $\mathbb{S}^3.$
This is a more sensible (IMHO) restatement of this question:
Which Compact $3$-manifolds with boundaries embed in $\mathbb{S}^3?$ Is there any hope of a characterization?
- 96.4k
20
votes
4
answers
3k
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Simply-connected rational homology spheres
Every simply-connected rational homology sphere is, in fact, the usual sphere in dimensions $2, 3.$ Is this true in dimension 4? Where are the first counterexamples? (I know there are some in dimensio …
- 96.4k
4
votes
1
answer
504
views
Quotient of principal congruence subgroups
This is a direct follow-up to this question. What is the quotient $\Gamma(2)/\Gamma(2^n)?$ (the principal congruence subgroups are in $SL(2, \mathbb{Z}).$ It is a 2-group, but what else?
- 96.4k
8
votes
1
answer
511
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Casson invariant
Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does …
- 96.4k
8
votes
1
answer
181
views
Hakenness of Heegard splitting
This is somewhat related to this previous quesiton. Suppose I give you a Heegard splitting of $M^3$ of genus $g$ with a gluing map $\phi.$ Is there some condition on $\phi$ which would guarantee that …
- 96.4k
5
votes
0
answers
244
views
Does every null-homologous surface bound, part deux
This is a follow-up to this question, and Mark Grant's excellent answer. The answer shows that the answer is yes, but what if one were under a strange compunction to actually construct the submanifold …
- 96.4k
11
votes
1
answer
327
views
Rank and hyperbolic volume
Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of Cu …
- 96.4k
9
votes
1
answer
509
views
Hyperbolic 3-manifolds fibering over the circle
Suppose you have the mapping torus $M_\phi$ of some pseudo-Anosov map $\phi.$ Is there some sufficient or necessary condition on $\phi$ to assure that $M_\phi$ has large injectivity radius? I am aware …
- 96.4k
5
votes
0
answers
142
views
List of cubical spaces
Suppose I have a three-dimensional cube (I tend to think of it as a regular ideal cube in $\mathbb{H}^3,$ but you don't have to). I glue up its sides in some way to obtain topological spaces. The ques …
- 96.4k