For questions on knot theory, the study of mathematical knots
1
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0answers
39 views
How to use two number to form a Jones polynomial
According to the Wikipedia article on Knots,
The number of crossing (rule $1$) and a line crossing the triangle (rule $2$) form a number such as $3,1$. With these two numbers, how do you form a ...
2
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1answer
50 views
Annulus Theorem
I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is
Use the Schoenflies theorem ...
1
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1answer
30 views
Uniqueness of Seifert graphs
If we make the bands and disks of a Seifert surface really small and really thin the surface collapses to a graph. It is called a Seifert graph.
If it is not a directed and weighted graph, can we ...
1
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1answer
40 views
Graphs from Seifert surfaces
Given a Seifert surface if we make the disks and bands infinitely small and thin it becomes a graph where the disks are vertices and the bands are edges. Can we say that following theorem,
For ...
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2answers
46 views
Uniqueness of Seifert surfaces of knots
I know the theorem that Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. So, can we also say that for every unique Seifert surface there is an unique knot and vice ...
3
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0answers
27 views
Alexanderpolynomial of connected sum via Fox calculus and Wirtinger presentation
Hello :) i have just reading the question "How to compute the Alexander polynomial of general torus knot" and i was suprised how strong it works if someone have a difficult question. I am also very ...
2
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0answers
28 views
Alexanderpolynomial of torus knot
i want to compute the Alexanderpolynomial of the torus knot $T_{p,q}$ with $p$ and $q$ coprime. I should work with the groups presentation $G(T_{p,q})=<x,y:x^p=y^q>$ of $T_{p,q}$. I have to use ...
1
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1answer
38 views
In topological terms, how would you describe the relationship between two consecutive links of a chain?
Consider the two rings that this magician is holding in his hands:
How would you describe that configuration in topological terms?
From a knot-theory standpoint, I would say that the rings form a ...
2
votes
1answer
47 views
Trefoil knot and Figure 8 knot are prime knots
I know that in general, it is difficult to tell whether a knot is prime or not. However, the Wikipedia page has established that the trefoil knot and the figure 8 knot are prime knots.
I've managed ...
0
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0answers
18 views
How to visulize surface link in four dimension?
I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...
4
votes
1answer
46 views
How to make a $C^1$ knot into a $C^\infty$ knot
Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
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1answer
20 views
Burau matrix of braid
What is the definition of a Burau matrix of a braid? Where can I find a definition?
Thanks
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0answers
19 views
Braid invariants resource
What are some Braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?
Thanks
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0answers
21 views
Knot quandle homomorphism
If you have a map that sends surjectively the generators of a knot quandle $\langle x_{i} , \ldots , x_{m} \mid r_{i} (x_{1} , \ldots , x_{m} ) \rangle$ to $\langle y_{i} , \ldots , y_{m} \mid s_{i} ...
1
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0answers
16 views
Visibility of symmetry in a link diagram
In "The First 1,701,936 Knots" it says that "any symmetry of a prime alternating link must be visible, up to flypes, in any alternating diagram of the link."
What is the formal definition of the ...
1
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0answers
24 views
seek visual pictures or video on decomposition of manifolds
In my study of knot theory, I notice that I lack examples to show some classical decomposition theorems in 3-dimensional manifolds, such as JSJ decomposition theorem, Milnor's prime decomposition ...
6
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0answers
79 views
Visualize Fourth Homotopy Group of $S^2$
I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
4
votes
1answer
26 views
boundary map in the (M-V) sequence
Let $K\subset S^3$ be a knot, $N(K)$ be a tubular neighborhood of $K$ in $S^3$, $M_K$ to be the exterior of $K$ in $S^3$, i.e., $M_K=S^3-\text{interior of }{N(K)}$.
Now, it is clear that $\partial ...
1
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1answer
20 views
Analogous notion of knot complements for braids
Knots/links seem to be studied quite a lot for their topological connection to 3-manifolds by considering knot complements in $S^{3}$. Is there an analogous topological entity for braids? They appear ...
0
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1answer
37 views
Is it true that any two tame knots are homotopic?
My understanding is that if the embeddings $f_0,f_1$ are tame knots then
$H(t,\theta) = (1-t)f_0(\theta) + t f_1(\theta)$
is a homotopy between them, thus all tame knots are homotopic. Is this the ...
3
votes
0answers
35 views
some help on the group of unknotted
Show that the group of the unknotted $K=\{(z_z,z_2)\in \mathbb{S^3} : |z_1|=1 \}$ is infinite cyclic. where $\mathbb{S^3}$ is to be considering as the unit vectors in $\mathbb{C^2}\cong \mathbb{R^4}$.
...
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0answers
23 views
How do Reidemeister moves affect Dehn presentations of knots?
How do Reidemeister moves affect Dehn presentations of knot groups?
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1answer
46 views
Types of knot invariants
Knot invariants seem to roughly be either numbers (that is, an amount of something ), polynomials, matrices, or groups. Are there any other invariants that have been studied that are not of this form?
...
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0answers
94 views
Conditions for a projection of a Knot to be a Knot diagram.
friends.
I'm working on a problem, the broad scope of which is to show that given a map $f:S^1\rightarrow \mathbb{R}^3$ be a smooth embedding, and a projection map $\pi_v:S^2\rightarrow P_v$, where ...
2
votes
1answer
69 views
An introduction to Khovanov homology, Heegaard-Floer homology
I am interested in knot theory and low dimensional topology. I would like to start studying Khovanov homology and Heegaard-Floer homology.
I (partially) read the original paper of Khovanov and then ...
1
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0answers
31 views
Skein relation described in terms of the planar algebra of tangles
Wikipedia says that "More formally, a skein relation can be thought of as defining the kernel of a quotient map from the planar algebra of tangles."
Does anyone know of a resource that further ...
0
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0answers
12 views
Gauss codes of isotopic reduced knots
Is there a way to go from one Gauss code of a reduced (minimum number of crossings) knot diagram to a Gauss code of an isotopic reduced knot diagram?
0
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0answers
54 views
Figure eight knot is not a torus knot
After seeing this picture of the figure eight knot:
Why isn't the figure eight knot considered a $(2,3)$-torus knot?
1
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1answer
32 views
Chessboard coloring of a knot
To construct a surface which boundary is a knot we can use the Seifert-algorithm. But we can also make a chessboard coloring of the knotdiagram $D$ of the knot $K$. So we get also an surface with ...
1
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0answers
45 views
Hamiltonian of one and two unknots
Recently I calculated the Ising Hamiltonian of a Hopf link. First, I colored the Hopf link in a checker board pattern and drew the Seifert surface from it. Considering the shaded regions as vertices ...
2
votes
0answers
20 views
Seifert surface and crossing number
i am sitting here with the problem of Seifert Surfaces. I know from a theorem that every knot does have a Seifert surface. We can also make a so called disc-and-band surface $F$ by gluing $v$ discs ...
1
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1answer
35 views
Are there any combinatorial studies of Kirby calculus?
All of the other diagrammatic calculi I know of can be utilised with basically just combinatorial knowledge - for instance calculating knot and link polynomials. Are there similar combinatorial ...
0
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0answers
19 views
Alexander polynomial calculation
With the original method of calculating the Alexander polynomial (see http://homepages.math.uic.edu/~kauffman/Alex.pdf), does it matter in what order the regions are labelled?
1
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1answer
32 views
A question about orthogonal projections of Smooth Embeddings of the circle.
Question: Let $f$ be a smooth embedding of $S^1\rightarrow \mathbb{R}^3$. Given an element $v\in S^2$ we have the orthogonal projection $\pi_v:\mathbb{R}^3\rightarrow P_v$ to the plane
$P_v$ = the ...
1
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0answers
33 views
Untieable knotted surfaces
After having seen how - comparatively - easy it is to untie a seemingly knotted surface embedded in $\mathbb{R}^3$, I am now looking for really (= untieable) knotted surfaces.
Is there a most ...
4
votes
2answers
156 views
A puzzle on knotted surfaces
Only after having learned that the somehow only notion of equivalence of knots is definitely "ambient isotopy" I stumbled over this blog entry on ambient isotopy. (Had it been earlier!)
What bothers ...
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1answer
117 views
Equivalence of knots
It's intuitively clear what it means that two knots $K,K'$ are essentially the same, but it can be termed and defined more precisely in different ways. Are all of them equivalent?
$K, K'$ are ...
22
votes
3answers
319 views
Picture of a 4D knot
A knot is a way to put a circle into 3-space $S^1 \to \mathbb R^3$ and these are often visualized as 2D knot diagrams.
Can anyone show me a diagram of a nontrivial knotted sphere $S^2 \to \mathbb ...
5
votes
0answers
75 views
Knots and graphs
Every knot gives rise to a number of 4-regular planar graphs - by regular projections onto the plane - which just have to be enriched by an over/under flag for every vertex to be able to reconstruct ...
3
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0answers
29 views
does a method exist to distinguish two component link consisting of just two unknots from an unlink?
Clearly, linking number is not enough as there are links like whitehead.
There is the enhanced linking number based on conway polynomial that can distinguish whitehead (and infinite family of such ...
2
votes
1answer
37 views
Concordant Links have Homotopy Equivalent Complements
Say I have a pair of links $L_0,L_1\subset S^3$ and an embedding $F:L_0\times I \rightarrow S^n$ such that $F(L_0,0) = L_0$ and $F(L_0,1)=L_1$ ($F$ is a concordance). Intuitively, the complements ...
2
votes
0answers
23 views
$3\tau(K_1$#$K_2)$=$\tau(K_1)\tau(K_2)$
Suppose we have two knots $K_1$ and $K_2$. Then look to the connected sum of $K_1$ and $K_2$ denoted by $K_1$#$K_2$ (defined for knots). Suppose $\tau$ is the number of $3$-colourings (definition for ...
3
votes
3answers
53 views
Differential characterization of unknots
How can the closed simple curves in $\mathbb{R}^3$ be characterized that can be boundaries of a 2-dimensional oriented surface in $\mathbb{R}^3$? Intuitively I would tend to say that it's exactly the ...
1
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0answers
46 views
Linking number and the factor $\frac{1}{2}$
i have a question about the linking number of a knot. Per definition: The Total Linking Number Lk(D) is obtained by taking half sum over all crossings (for more definition look to other definitions ...
1
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0answers
42 views
Torus link and knots
Hello :) i am reading about knot theory especially torus links :) i read "Crossing number and Torus links" and the answer isn't clear. Does there exist a solution without topology but with group ...
1
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1answer
68 views
How is PL knot theory related to smooth knot theory?
I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
1
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1answer
52 views
Why is there no contradiction by construction of alternating knots? [duplicate]
I have got a question. From definition alternating diagram $D$ of a knot $K$ is a diagram such passes alternately over and under crossings. A knot $K$ with such a diagram $D$ is called a alternating ...
0
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0answers
31 views
Signature of a Torus Link
The signature of a torus knot $T_{p,q}$, where $p$ and $q$ are coprime, is well-defined and relatively easy to compute in terms of lattice points in certain quadrilaterals, as a summation over floor ...
0
votes
1answer
52 views
Crossing number and Torus links
We define the crossing number of a knot $K$ to be the minimal number of crossings in any diagram of $K$. Surely we can easy prove that there do not exist knots with crossing number $1$ and $2$ ...
2
votes
1answer
81 views
Isotopy and Homotopy
What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.
