For questions on knot theory, the study of mathematical knots

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3
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1answer
18 views

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 ...
2
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0answers
55 views

Surgery to unlink $S^1$ and $S^2$ in $S^4$

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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0answers
30 views

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: ...
2
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0answers
16 views

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 $-K$ ...
1
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1answer
28 views

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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0answers
11 views

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 ...
3
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1answer
22 views

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 ...
4
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0answers
61 views

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 $\mathbb{R}^3$?". ...
2
votes
1answer
36 views

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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0answers
15 views

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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0answers
33 views

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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0answers
36 views

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 ...
0
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0answers
22 views

Skein relationship and Alexander polynomial

Given is the three link diagrams of Conway $L_{0},L_{+},L_{-}$ and the corresponding Seifert matrices $M_{0},M_{+},M_{-}$. Prove that ...
0
votes
1answer
23 views

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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0answers
17 views

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. ...
0
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1answer
27 views

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 ...
1
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1answer
36 views

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?
3
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0answers
125 views

Generators of the braid group

Let $C$ be the plane curve in $\mathbb{C}^2$ defined as $\{x²-y^3=0 \}$. The fundamental group of $\mathbb{C}^2 \backslash C$ is the same of the trefoil knot : $\langle a_0,a_1 \; : \; a_0a_1a_0= ...
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0answers
20 views

Proof of the existence of knot which realizes given polynomial as Alexander poly.

The following theorem by Seifert is well known. A first half of the proof can be found in the book by Rolfsen (pp.171-172). I have questions that can be asked along the way, but the main question is ...
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0answers
7 views

t-minimal diagram of a surface in 4-space

Let D be a t-minimal surface diagram of a surface embedded in 4-space, i.e. D has minimal number of triple points over all possible surface diagrams. Is it true any double curve in D contains all ...
2
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0answers
29 views

Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For ...
0
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0answers
22 views

smoothing crossing point in knot diagram

For an oriented knot diagram, each crossing can be smoothed in two possible ways (we trace along one strand and before the crossing we move to the other strand) an oriented smoothing and an ...
2
votes
1answer
29 views

genus of knot and crossing number invariants

Genus of knot is defined to be the least genus among all Seifert surfaces of knot. Crossing number is the minimal number of crossings over all possible diagrams. Both genus of knot and crossing ...
2
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1answer
40 views

additivity of crossing number of composite knots

Let $K_1 \# K_2$ denotes the connected sum of tow knots $K_1$ and $K_2$. The crossing number is the minimal number of crossings among all knot diagrams, denoted by $c(K)$. It is conjectured that ...
3
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0answers
40 views

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 ...
2
votes
1answer
151 views

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 ...
0
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1answer
24 views

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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0answers
53 views

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 $S^1$ ...
2
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0answers
43 views

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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0answers
25 views

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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0answers
20 views

Smoothing multiple crossings

Can you smooth multiple crossings at once in a knot when computing the bracket polynomial? If so is a there a rule for this?
0
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1answer
24 views

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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0answers
14 views

Linking number and writhe of the hopflink

If you have two circles linked up to give the hopflink. Both these circles have clockwise orientation. Is the linking number and writhe going to be zero?
0
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1answer
35 views

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?
10
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5answers
359 views

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 ...
0
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1answer
31 views

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?
0
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0answers
19 views

Compatible seifert circles

When are two seifert circles compatible? I originally thought that it was just a matter of looking at their orientations to check but it seems that is not enough.
1
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0answers
27 views

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 ...
2
votes
1answer
70 views

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 ...
0
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0answers
43 views

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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2answers
36 views

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?
3
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1answer
80 views

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?
1
vote
1answer
46 views

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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0answers
63 views

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 ...
2
votes
1answer
32 views

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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0answers
16 views

unknotting number of surface-knot

Unknotting number of surface-knot is an invariant which is defined by the minimal number of 1-handles attached to obtain an unknotted over all possible surface diagrams representing it. A surface ...
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0answers
30 views

The HOMFLY polynomial existence proof

Theorem: There is a function \begin{align*} P:\{\text{Oriented links in $S^4$}\}\rightarrow\mathbb{Z}[l^{\pm{1}},m^{\pm{1}}], \end{align*} which satisfies the three axioms for the HOMFLY polynomial. ...
2
votes
1answer
57 views

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 ...
2
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0answers
33 views

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 ...
8
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2answers
132 views

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 ...