# Questions tagged [low-dimensional-topology]

Low-dimensional topology generally refers to the study of 3 or 4 dimensional topological manifolds and knot theory.

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### Help with moduli spaces of four-marked spheres

I'm thinking about the moduli space of the four-punctured sphere where some of the removed points are distinguishable and some are indistinguishable. I believe there should be some covering maps ...
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### Finiteness in Prime Decomposition Theorem for 3 manifolds

In Allen Hatcher's text on 3 manifolds, he proves the Prime Decomposition Theorem by showing that a collection S of 2 spheres embedded in a smooth and compact 3-manifold M satisfying the condition ...
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### Notes on Low-Dimensional Topology

I am studying algebraic topology at the moment and I'm halfway done with Hatcher's book. I am extremely interested in low-dimensional topology, so I was wondering if anybody knows a good set of notes ...
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### Understand the topology of this 4-dimensional data.

Now I have a set of 4 dimensional data points. If I project them onto the first and second dimension, they look like figure 1. If I project them onto the third and forth dimension, they look like ...
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### Neighbourhood retract of a loop

Suppos $f:S^1\to M$ is a smooth map and $M$ is a smooth manifold. Does there exist a neighborhood $\mathcal N$ of the image $f(S^1)$ which retract (or deformation retract) to $f(S^1)$? If so, is there ...
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### References on hyperbolic geometry and Teichmuller Theory

I am asking a soft question here. I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
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### The 3-component hopf link covering the trefoil

Let $H$ be the complement of the 3-component Hopf link, which is homeomorphic to the complement of the 3-chain link if you prefer, and let $T$ be the trefoil knot complement. I recently encountered ...
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### Recommendation for Low-dimensional topology textbook [closed]

Can anyone here recommend a low-dimensional topology textbook that contains knot theory and 3,4-manifolds?Or should I look for these subjects in separate textbooks?
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### Reference request for 3-manifold

I am asking a soft question. I am planning to learn $3$-manifold using the book "Geometry and topology of three-manifolds" by William Thurston. I want to know how much of Riemannian geometry,...
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### What are the dimensions of a 4D cube represented in 3D?

I'm hoping to construct a physical model of a 4D cube. However, I'm struggling to work out the proper size ratio between the inner and outer cubes. From the graphics I've seen, the inner cube seems ...
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### Triangulation that includes given subcomplex

I was reading Prasolov and Sossinsky's book Knots, Links, Braids and Three-Manifolds and came across the following statement in the proof of Theorem 9.2: We can assume $L^3$ has a triangulation $K$ ...
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### Topological name of nearly-reversed torus

I have a mountain floating in space. Topologically, it's a sphere. I put a cave in the mountain. Topologically, it's still a sphere. I add a stalactite (a cone of rock hanging from the roof of the ...
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### Double of a drilled out hyperbolic 3-manifold

If one starts with a closed hyperbolic 3-manifold and removes the interior of a some number of solid tori. The resulting manifold with boundary can be doubled along its boundary. Does this doubling ...
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### Equivalent bridge representations of a link

I'm looking for something like an analogue of Markov's theorem (which states necessary and sufficient conditions for two braids to be closure equivalent) in the context of bridge representations of ...
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### Compact 3-manifold with compressible boundary tori?

There is much to be said about 3-manifolds with boundary consisting of a possibly empty collection of incompressible tori. I don't seem to know where to look to find much about 3-manifolds with ...
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### Relationship between ends of a hyperbolic 3-manifold and ideal boundary components

I've been reading Sullivan's paper "Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups", and I was hoping I could get some ...
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### Realization of subgroups of $\pi_1(M)$ by submanifolds

Let $M$ be a smooth oriented connected 3-manifold, let $G$ be the fundamental group $\pi_1(M)$. If $M$ is compact, by Kneser-Milnor Decomposition Theorem $M$ can be written as a connected sum of prime ...
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### Haken's Algorithm for Unknot Recognition

I'm an undergrad studying knot theory and I'm exploring the problem of unknot recognition. I believe I understand Haken's algorithm at a high level but I'm having trouble understanding why it is ...
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### What does the commutator subgroup of a knot group “look like”?

For $X = S^3 - K$ a knot complement, the abelianization $$1 \to [\pi_1(X),\pi_1(X)] \to \pi_1(X) \to \mathbb{Z} \to 1$$ has a right split, where $\mathbb{Z} \cong \pi_1(X)^{ab}$ is realized by a ...
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### Can we see both sides of faces of 4D tesseract from 3D?

To see the whole surface of a cube it is sufficient to have two viewpoints above two opposite vertices of a cube. We can see three faces of a cube from the first viewpoint and other three faces from ...
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### Are topological 1-manifolds embedded in 2-manifolds always locally flat?

I would like to know whether the following is true: Let $M$ be a topological 2-manifold (without boundary), and let $i: [0,1] \to M$ be a continuous embedding. Then $\mathrm{Im}(i)$ is locally flat in ...
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### What is the point of the differing appearances of the HOMFLY skein?

What is the point of the differing appearances of the HOMFLY skein? For example, can somebody explain to me why one may prefer the relation $tP_P(q,t)-t^{-1}P_-(q,t)=(q-q^{-1})P_0(q,t)$ to the ...
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### Are these two links equivalent?

Are the following two links equivalent (orientation preserving isotopies)? The two links have the same linking number. The only difference is the crossing that in one case is positive while in the ...
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### The Seifert form of a fibered knot.

Is the Seifert form of a fibered knot always unimodular? It is known that $H_1 (F)$ $\cong$ $H_1 (S^3- F)$ for any knot. Probably we have to use the Alexander duality, but I don't see how.
I know that there is this result of Milnor that all algebraic links are fibered. And the $(p,q)$-torus link is an algebraic link. But then I'm reading this paper of Baader and Graf (http://dx.doi.org/...