# Tagged Questions

For questions on knot theory, the study of mathematical knots

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### Showing that gluing two knot exteriors together contains subgroups isomorphic with the knot groups.

I'm working through Rolfsen's "Knots and Links" and section 9D exercise 10 has me stumped: Let $K_1$ and $K_2$ be knots in two separate copies of $S^3$ with respective meridians $m_1$ and $m_2$ and ...
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### Definitions from topology

I'm reading some papers on the unknotting problem in Knot theory and am running into some notation I don't know (my exposure to topology is minimal, but I have seen it in Analysis courses, Algebra, ...
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### Books about braid theory

I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. Thanks for any ...
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### Background required to understand the mathematical definition of knots and their transformations

What are the concepts of math required as a prerequisite to understand Knot Theory? I'd like to be able to make a humble beginning by being able to mathematically define knots and the non-rigid ...
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### What is a 3-cell? What is a 3-disk?

Checking the usual places on the Web doesn’t (right now) yield a short answer to this simple question. I’m worried that I will spend just as long trying not to confuse $n$- with $(n+1)$-cells as I did ...
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### Can a trefoil knot be stretched to look like a triangle with three knots at the vertices?

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices, like in the right side of the image below, or is that transformation impossible to happen? If possible, what ...
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### How to prove a knot with genus larger than 1 is prime, such as Miller Institute Knot?

It is easy to show that a knot with genus 1 is a prime knot because the genus is additive under direct sum. However, I found that some prime knot, for example, $6_2$ the Miller Institute Knot have ...
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### Surgery to unlink $S^1$ and $S^2$ in $S^4$ [closed]

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
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### Linking of $S^p$ and $S^q$ in the $\mathbb{R}^d$ space

Can we have a nontrivial linking of a $S^p$ sphere and a $S^q$ sphere in the $\mathbb{R}^d$ space (or in the ${S}^d$ space)? I suppose that it can happen only if $p+q<d$. For example, we can have: ...
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### colouring knot diagram and its mirror by the same colouring

Let $K$ be a knot diagram coloured by any quandle $X$. Let the colouring used be $C$. Reverse the orientation of $K$ to obtain the reverse of $K$, denoted by $-K$. Then is it possible to colour $-K$ ...
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### 1-surgery on the figure-eight knot: reference request

As far as I know, 1-surgery on the figure-eight knot gives ($\pm$) the Brieskorn sphere $\Sigma(2,3,7)$. However, is there a citeable source for this? Sometimes Thurston's notes are mentioned, but I ...
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### Recovering knot crossing orientations from a Gauss code or Dowker notation

Some common representations of knots do not directly give the sign/orientation of each crossing. For instance, the trefoil knot has Gauss code -1, 3, -2, 1, -3, 2 and Dowker-Thistlethwaite code 4 ...
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### property of quandle

Let $Q$ be any quandle. Let $y,k, w$ be elements of $Q$. Is it true that if $y*k=y*w$ then $k=w$? I don't think so since the second axiom of the quandles states that for any two distinct ...
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### Are there enough knots to cover $\mathbb{R}^3$? [closed]

Actually, several years ago I was in a short, introductory, course about knot theory, and my original question that I posed was: "can the knots be used to classify homeomorphims in $\mathbb{R}^3$?". ...
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### Are knot complements prime 3-manifolds?

Well, that is basically the question. Is the complement of a knot, i.e. an smoothly embedded copy of $S^1$ in $S^3$ a prime $3$-manifold? Here I mean by prime: A connected $3$-manifold $M$ is prime ...
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### unknotted $n$-dimensional knot

Let $n$ be any integer. An $n$-dimensional knot is an $n$-dimensional manifold embedded smoothly into $\mathbb{R}^{n+2}$. If it is homeomorphic to a disjoint union of $n$-spheres, then it is denoted ...
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### Determining slice knots

Lately I have been thinking about slice knots. Is there any known effective procedure for determining whether a knot is a slice knot?
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### Seeking guide for project.

I have to submit a project within 2 months for 4th semester(M.Sc). I wish to do it on knot theory, although I know little about it. My plan is to make it an introduction to the subject and to ...
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### Does the elementary knot move really preserve the orientation?

So in this picture, the first diagram changed to the third diagram by the elementary knot moves, but the orientations of the first and the third are different. I wonder if in $R^3$ the moves don't ...
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### Is the Alexander Ideal of a Link always Principal?

It is known that the Alexander ideal of a knot (i.e., a link of one component) is always a principal ideal since any tame knot in $S^3$ has a square presentation (Rolfsen, D. Knots and Links, pp. 206-...
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### Is the Ambient Isotopy relation (for embeddings) symmetric?

Definition: Let $X$ and $Y$ be topological spaces. Suppose that $Y$ is compact and Hausdorff. Let $f,g:X\to Y$ be embeddings. We say that $f$ is ambient isotopic to $g$ (denote $f\sim g$) if there is ...
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### Is the closure of $[\sigma_1^2,\sigma_2^2]$ in $B_3$ equal to the Borromean rings?

Is the closure of $[\sigma_1^2,\sigma_2^2]\in B_3$ (the braid group with $3$ strings) equal to the Borromean rings? If yes, is there any simple proof?
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### What is the relation between the construction of Reshetikhin-Turaev and Witten's paper?

I have read that the papers of Reshetikhin and Turaev provide a mathematically rigorous framework for the ideas contained in Witten's paper "Quantum field theory and the Jones polynomial". Can ...
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### Software for drawing braid-related graphs

I am looking for a (preferably free) software that can draw braid-related graphs, such as and (Quoted from A Study of Braids By Kunio Murasugi, B. Kurpita) I have seen this question, but the ...
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### How to construct a concrete isotopy from figure eight knot to its mirror image?

We know that figure eight knot is amphicheiral. But how to show that in terms of Reidemeister moves? Just several instructive hints will help a lot. I really have no idea..
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### Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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### if a circle is deformation retract of a manifold; then is there a knot deformation retract to an embedded M in 3-space

Let M be a metric space such that the circle is deformation retract of M. If a circle is embedded in 3-space, we obtain a knot. Suppose the space M is embedded in 3-space and its projection image in ...
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### fundamental group of ribbon knot

The ribbon knot is a boundary of self intersecting disk such that the singularity is a self intersection of the disk along an arc. My question is that how to compute the fundamental group of ribbon ...
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### Bracket polynomial of a hopflink

When you are computing the bracket polynomial of the (for example) hopflink, why can you not smooth all the crossings in one go? Why do you have to only first start with one crossing? For example in ...
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### Reidemeister infinity move

If you have a knot K. How do you apply a R infinity move to it? Is their an algorithm which tells you what to do at certain parts on a knot?
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### An Illustrated Classification of Knots.

Let me be honest here: I know very little about Knot Theory. I'm sorry. I've a friend though, someone with no training in Mathematics at all but who is a huge fan of knots (for whatever reason), who ...
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### embedding projective plane in 4-space? [closed]

Is it possible to embed projective plane in 4-space? If not what is the reason and what is the smallest singularity set?
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### Is there a complete Link Invariant for links with N crossing.

Are there known examples of pairs $\left(f, N\right)$, where $f$ is a link invariant that is known to be complete when restricted to link diagrams that have at most $N$ crossings? (Ideally, f should ...
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### math in horseshoe puzzle

We know that Rubik's Cube is a good demonstration of group theory. Correspondingly, for the horseshoe puzzle as in the picture below, is there a math language for it? Does it demonstrate any math ...
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### Is it trivial that you cannot get apart 2 linked unknots?

I have read that: a trefoil is not an unknot is not trivial in knot theory. And it put the following question in me. Is it also not trivial that you cannot get apart 2 linked unknots? I mean, is it ...
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### Knots isotopic to their mirror image

How do you prove a knot is achiral? Do you just show swap the under crossings and the over crossings?
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### Is there any knot showing infinitely many crossings?

Let $K$ be any unknot. Is it possible that $K$ shows infinitely many crossings? And if it is possible: How to get $K$ from the simplest unknot through Reidemeister moves?
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### Are there knots that can be distinguished between by the Jones polynomial that the HOMFLY polynomial cannot distiguish?

Obviously the HOMFLY polynomial is a more superior invariant than the Jones polynomial since it is a geralisation of the Jones. However, are there knots that can be distinguished between by the Jones ...
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### What is the most rigorous book for knot theory?

I have read the knot book by Colin Adams and it has been very helpful with getting to grips with the basics. I struggle with showing things and proving things. I need a book with a lot of rigorous ...
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### Knots winded around surfaces of a higher genus

There are many results for the knots winded around a torus, but what about knots winded around surfaces of a higher genus? Is there any classification of such knots? I would be glad to see any review ...
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### Isotopic tori in $\mathbb{R}^4$

Intuitively it seems to me that two tori in $\mathbb{R}^4$ are isotopic to each other. By isotopic, I mean a smooth family of deformations beginning in one and ending in the other, and each member of ...
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### Is there a Knot Theory software to analyze general curves in 3D?

So I happen to like proteins quite a lot and one thing that is very similar to a protein, when represented as the bare minimum, is a 1D curve embedded in the 3D space. They form beautiful and unique ...
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### Jones Polynomial from Statistical Mechanics

I've been told that, given a knot projection, there is a way of associating a statistical system in such a way that the partition function of the system corresponds to the Jones polynomial of the ...
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### Are Knots closed?

In every definition I see, a (classical) knot is an embedding of $S^1$ in $S^3$ or $\mathbb{R}^3$. But my lecturer said that the complement of a knot in $S^3$ is open, hence the knot is closed. But I'...