# Questions tagged [3-manifolds]

A 3-manifold is a space that locally looks like Euclidean 3-dimensional space

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### Reference request: A knot is tame if and only if it has a tubular neighbourhood

Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...

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### A coincidence between the Lambert cube, Lobell polyhedron, and hyperbolic 3-manifolds?

I. Lambert cube $\mathfrak L(\alpha_1,\alpha_2,\alpha_3)$
In this paper (p.8), we find the volume $V$ of the hyperbolic Lambert cube for the special case $\alpha=\alpha_1 = \alpha_2 = \alpha_3$ as
$...

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### Does WRT invariant detect hyperelliptic involution on the genus 2 surface?

The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\...

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### Obtaining the bounding 4-manifold from the Heegaard diagram

It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...

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### Prerequisites for Hempel's 3 manifolds?

What are the prerequisites for Hempel's 3 manifolds book? I've heard you need a good amount of PL topology. I have read Jennifer Schulten's book and found it accessible, but I've heard Hempel is tough....

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### Are Turaev-Viro invariants holonomic?

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...

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### Counting fixed points for Hamiltonian symplectomorphisms on $T^{2}$

This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in ...

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### Quotient of Three Dimensional Torus by Permutation on Coordinates

The Mobius Strip can be realized as a quotient of $T = (S^1)^2$ via the identifications $(x,y) \sim (y,x)$.
I tried to generalized this concept to a higher dimension, and consider the quotient of $(...

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### Are triangulations with common refinements PL-homeomorphic?

Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...

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### What are these 3-manifolds from surgery?

I know that surgery on the unlink with +0 slope gives $S^2 \times S^1$ (where all the links above are embedded in $S^3$). I figured (I think) that surgery on the hopf link (with +0 on both) returns $S^...

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### Circle bundles and surface bundles which admit no strongly irreducible Heegaard splittings

Let $S$ be a closed connected orientable surface with $g(S)>0$. Jennifer Schultens, in her paper ``The Classification of Heegaard Splittings for (Compact Orientable Surface)$\times S^1$'', proves ...

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### SnapPy isometry routine

Dear Colleagues and Friends,
Here's a question that I hope some of you, more experienced in programming, can answer.
Once SnapPy is used to compute the symmetry group of a hyperbolic manifold by ...

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### Existence of smooth structures on topological $3$-manifolds with boundary

It is said in this thread Unique smooth structure on $3$-manifolds that every topological $3$-manifold admits a smooth structure. However it is not specified whether the manifolds are allowed to have ...

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### open book decompositions of $\Sigma\times S^1$

Let $\Sigma$ be a closed orientable surface. Is there a standard open book decomposition on the $3$-manifold $M=\Sigma\times S^1$?
If the binding is allowed to be empty in the definition of an open ...

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### 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 ...