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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Do the link cobordism induced maps on Khovanov homology for different diagrams commute?

I'm trying to determine the validity of a property of the morphisms on Khovanov homology that are induced by oriented link cobordisms. These maps are defined with fixed diagrams for the boundary links,...
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1answer
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Orbits of points under pseudo-Anosov diffeomorphisms

I have no intuition about orbits of pseudo-Anosov diffeomorphisms $\phi$ of closed surfaces $S$ of genus $>1.$ I understand that there are infinitely countably many periodic points, correct? What ...
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Distortion of the Unknot

In Mikhail Gromov's "Filling Riemannian Manifolds" he defines the distortion of a knot $K$ embedded in $S^3$ as $$\delta (K) := \inf_{\gamma \in K} \sup_{x,y \in \gamma} \frac{d_{\gamma}(x,y)}{||x-y||}...
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Seifert Fibered Spaces with Boundary are $\mathbb{P}^2$-irreducible

I'm reading Peter Scott's The Geometry of 3-Manifolds and am trying to understand the argument behind this statement, which arises in the proof of Corollary 3.3: If $M$ is a Seifert fibered 3-...
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1answer
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Tying a portion of a knot diagram

I am having some trouble with a paper by Livingston (Infinite Order Amphicheiral Knots, Algebraic and Geometric Topology 1, 2001, 231-241). He starts with a knot $K$, and constructs a new knot "...
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A knot $K = K_1 \# K_2$ is alternating if and only if $K_1$ and $K_2$ are alternating.

Is a knot $K = K_1 \# K_2$ is alternating if and only if $K_1$ and $K_2$ are alternating? In particular, I'm interested in the following direction: If $K_1 + K_2$ is an alternating knot, are both $...
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1answer
36 views

Is it easy to see that $S^2\times S^1$ does not admit Euclidean, Hyperbolic or Elliptic geometry?

It is easy to see that $S^2\times S^1$ as a Riemannian manifold is not Euclidean, hyperbolic or elliptic. Is it also easy to see that the topological manifold $S^2\times S^1$ does not admit one of ...
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How would we perceive space from within the hypersphere $S^3$?

Imagine being in a relatively small 3-sphere. Since geodesics behave differently from euclidean space, light will follow different trajectories, and I am wondering what the impact would be on our ...
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Do there exist non-trivial knots whose Jones polynomial is a unit?

Question: It's an open problem whether or not the Jones polynomial distinguishes the unknot from all other knots. That is, the following problem is unsolved. Does there exist a knot $K$ which is ...
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Incompressible surfaces in surface bundles of $S^1$ that are homologous to the fiber are homotopic to the fiber?

I was reading Thurston's "A norm for the homology of 3-manifolds" and I had some questions that I think are pretty basic but have me stumped at the moment. Let $M^3$ be a compact 3-manifold that ...
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Why is the universal cover of $SL(2,\mathbb{R})$ a line bundle over $\mathbb{H}^2$?

In Peter Scott's "The geometry of 3-manifolds" he says that: "As $U\mathbb{H}^2$ is a circle bundle over $\mathbb{H}^2$, we see that $\widetilde{SL(2,\mathbb{R})}$ is naturally a line bundle over $\...
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Why can't Antoine's necklace fall apart?

Antoine's necklace is an embedding of the Cantor set in $\mathbb{R}^3$ constructed by taking a torus, replacing it with a necklace of smaller interlinked tori lying inside it, replacing each smaller ...
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1answer
37 views

Reference request for surgery on knots

I've seen this article https://www.math.cuhk.edu.hk/~ztwu/JonesCosmetic.pdf on the Jones Polynomial and Cosmetic Surgery and I've looked at the Wikipedia entry on Dehn surgery as well. My background ...
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Is the linking number via Seifert surfaces well defined?

Let $i(K, F_L)$ be the signed count of intersections of an oriented knot $K$ with a Seifert surface $F_L$. (That is, $F_L$ is an oriented compact surface with boundary $L$ for some knot $L$.) I want ...
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Knots and Embedding

What is the different between a smooth embedding and a homeomorphism? I read a little about knots and I do not understand how a knot can be an embedding of the unit circle but not homeomorphic to it?
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Heegaard genus and ranks of fundamental groups

I found an interesting conjecture in the paper "Hyperbolic volume, Heegaard genus and ranks of groups" by Peter B. Shalen. It says that if $M$ is a compact, orientable, hyperbolic 3-manifold, then ...
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1answer
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Weeks Manifold - the Heegaard splitting and presentations of its fundamental group

I'm working on the Heegaard splitting of some 3-manifold and I'm currently stucked on the Weeks manifold. It is a closed orientable hyperbolic 3-manifold with the smallest volume, obtained by (5, 2) ...
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1answer
105 views

Can one piecewise function have two metric spaces? [closed]

Can one make a piecewise function on the manifold of complex numbers $\mathbb{C}^1$ that works on two metric spaces? A metric space is a set together with a metric, a means to tell the distance ...
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1answer
28 views

How can I find a medium-length word equal to 1 on a 2-torus?

I'm trying to give an example of Dehn's algorithm on a 2-torus and to do so I want to find a word about 20 letters or so in length that is equal to 1 so that I can apply the algorithm. I'm having ...
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1answer
33 views

Fundamental group of quotient of SO(3) by icosohedral group

I was recently reading an expository paper by Milnor and came across the following claim (in the footnote on the 2nd page): If $I$ is the subgroup of $SO3$ consisting of rotational symmetries of ...
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1answer
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Isomorphism between braid groups

Let M be a connected topological manifold of dimension $\geq2$ and let $M^n=M\times\dots\times M$ be the product on $n\geq1$ copies of M with the product topology. Set $\mathcal{F}_n(M)=\{(u_1,u_2,\...
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Understanding the proof of Scott's core theorem

I am having some trouble understanding the proof of Scott's core theorem as presented by Rubinstein and Swarup (link, requires academic access). Scott's core theorem states the following: if $M$ is a ...
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Homotopy equivalence in relative Diffeomorphism group

I am currently working on a project where one studies a smooth bordered compact $3$-manifold $M$ with some properly embedded essential surface(s) $S \subset M$. More precisely, I am interested in the ...
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26 views

Dimensionality of 1D line through a 2D plane

What is the dimensionality of the 2D plane with a 1D line going perpendicular through it as depicted here? Could one come up with a coordinate system that describes the entirety of this space? In ...
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The proof of the quotient space of a manifold with properly discontinuous action is an orbifold.

I am reading the proof of the following proposition from Thurston's the geometry and topology of 3-manifolds: My questions are: By $U_x=\tilde{U}_x/I_x$, if $\cap_{i=1}^kU_{x_i}\not=\emptyset$, then ...
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Irreducible $3$-manifolds with boundary

A $3$-manifold $M$ (without boundary) is irreducible if every smoothly embedded $2$-sphere in $M$ bounds a $3$-ball. What would be a “good” definition of irreducibility for $3$-manifolds with ...
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Describing the fibers of the (induced map on $\pi_1$) of the inclusion $\Sigma_{g,b}\to\Sigma_{g,0}$

Let $\Sigma_{2,1}$ denote the genus 2 orientable surface with a single boundary component. Then we have an inclusion $\Sigma_{2,1}\hookrightarrow\Sigma_{2,0}$ which induces a surjection on fundamental ...
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1answer
88 views

A wild knot and its complement

I have been reading about wild knots in $\mathbb{R^3}$ that have nonsimply connected complements. I'm a bit confused here and I added a picture of a wild curve with two circles in its complement. Is ...
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1answer
61 views

3-manifolds: connected sum & handle decomposition

Suppose we are given a 2 diagrams for an handle decomposition of two closed, orientable 3-manifold M and $\tilde{M}$. Each of these diagrams consists in an handlebody of genus $g$ (respectively $\...
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Equivalence of two elements in $\pi_2(BX)$, where $BX$ is the rack space corresponding to the quandle $X$.

I am reading the proof of Lemma 4.1 given in the chapter "Some of Quandle Cocycle Invariants of links" of the book "Quandles and Topological Pairs" by "Nosaka." Before coming to the question, I have ...
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1answer
39 views

Mapping class group of $S^1 \times S^1 \times I$

I am interested in the computation of the mapping class group of the manifold $M=S^1 \times S^1 \times I$. One can visualize $M$ as a "torus cross $I$", or as an "annulus cross $S^1$". $M$ is ...
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1answer
58 views

The 2 skeleton of a 3 manifold is the 2 skeleton of a $K(\pi, 1)$

Recall that a topological space $M$ is called aspherical if is path connected and the homotopy groups $\pi_n(M) $ vanish for $n\geq 2$. A (smooth) 3-manifold $M$ is an homology sphere if $H_*(M,\...
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1answer
65 views

Homotopy Type of Embeddings of Circles in a 3-manifold

I am wondering how one could compute the homotopy type (or weak homotopy type) of the embedding space which I describe now. I suspect it to be contractible but this is merely an intuition. Consider ...
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1answer
49 views

why is a loop on a flat torus unshrinkable?

I am reading Knows, Molecules, and the Universe: An Introduction to Topology. A definition of for shrinkable has been given (pg. 68, chapter 3): If a loop can be pulled in, we say the loop is ...
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1answer
23 views

Reference for basic examples of skein algebras

I’m aware of the magic of skein algebras of surfaces only recently. It connects to knot theory in several ways. But while there’s abundance of basic texts on knot theory, I cannot find some for skein ...
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Definition of compatible triples in Turaev's Quantum Invariants for knots and 3-manifolds

I'm currently reading Turaev's book "Quantum invariants..." and I'm struggling with the following definition on page 62. Where $A,B,C$ is one of the following crossings (up to isotopy) representing ...
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1answer
96 views

Tubular Neighborhoods vs. Regular Neighborhoods

I am trying to understand the difference between a tubular neighborhood and a regular neighborhood. I have a theorem I want to use that applies to regular neighborhoods, and I am unsure if it will ...
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1answer
89 views

Knot group is $\mathbb{Z}$ iff $K$ is the unknot

Let $K \subset S^3$ be a knot; we call knot group the fundamental group of the complementary of $K$ in $S^3$. I've come across the fact that the only knot whose knot group is isomorphic to $\mathbb{Z}...
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1answer
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Units of triangulation in general 3-manifold

A caveat to the following question is that this might be a super stupid question for topologists :). Sorry that I am quite new to the field. What I understand regarding triangulation for a surface (2-...
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Mapping Class Group of Pants with a Hole

It is known that the mapping class group of the torus $\mathbb{T}^2$ is $\text{Mod}(\mathbb{T}^2) \cong \text{SL}_2(\mathbb{Z})$. We also know that for a pair of pants $P$ (a sphere with three ...
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A factorization result for links

I am currently informing myself about the basics of knot and link theory. I learnt pretty quickly about prime knots and a unique decomposition theorem for knots stating : $\text{Thm: Every (isotopy ...
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1answer
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Classification of links with unknot components

Question I am interested in the links in $\mathbb{R}^3$ with trivial components. More precisely, I'd like to know if the classification of links with finitely many components, which are all unknots ...
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Given a $3$-manifold $M$, is $M\# S^3$ homeomorphic to $M$?

Given a surface $N$, we have $N\# S^2$ homeomorphic to $N$, I wonder if given a $3$-manifold $M$, we have $M\# S^3$ homeomorphic to $M$?
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1answer
25 views

Average property for the Fox n-coloring (knot theory)

For the Fox n-coloring (knot theory) there is the property that Around a crossing, the average of the colors of the undercrossing arcs equals the color of the overcrossing arc. This matches with my ...
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Explicit immersion of $RP^3$ in $R^4$

The existence of immersions of real projective 3-space in 4 dimensional Euclidean space was proved by Hirsch (I think) in 1959. I would like to see an explicit example of such an immersion. Does ...
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About a sub-manifold of $S^3$ whose boundary only consists of tori

I am reading a paper called "JSJ-decomposition of knot and link complements in $S^3$", written by Ryan Budney. My question does not concern the essence of the paper but a technical fact about 3-...
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1answer
33 views

Continuous functions from $(X,d)$ to $(\mathbb{R},\mid\cdot\mid)$, open set

If $f$,$g$ are continuous functions from $(X,d)$ to $(\mathbb{R},\mid\cdot\mid)$. Show that $A=\{x \in X : 2f(x) > 3g(x)\}$ is open in $X$. I understand that the continuous image of an open set is ...
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1answer
155 views

Why does $\mathbb{RP}^2$ not continuously embed in $\mathbb{R}^3$?

Ok, I know the answer: any closed hypersurface of $\mathbb R^3$ is orientable while $\mathbb R \mathbb P^2$ is not. But I know how to prove that only for smooth embeddings. Is there a simple way to ...
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1answer
67 views

Are enclosed 4D shapes really enclosed?

Imagine we have a 4D cube (hypercube) and a 3D sphere (sphere) inside it. The hypercube consists only of its 24 (2D square) faces. The question is: are there any possibilities for the sphere to escape ...
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277 views

A “double” Möbius Strip

A few days back, I was taught about equivalence relations, classes and was given some glimpses of quotient topology. Suppose we are given the space $[0,1]^{2} \subset \Bbb{R}^{2}.$ Then, the ...

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