For questions on knot theory, the study of mathematical knots

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9
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2answers
282 views

How to start learning knot theory?

Knot theory really sounds cool and I'm very interested in it. But I'm wondering what basic knowledge it is required and how I should start learning about it. Thanks
3
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1answer
150 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
59 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 ...
2
votes
1answer
60 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 ...
1
vote
2answers
129 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
votes
0answers
50 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
61 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
vote
1answer
66 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 ...
3
votes
1answer
183 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 ...
4
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1answer
66 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 ...
4
votes
2answers
99 views

Burau matrix of braid

What is the definition of a Burau matrix of a braid? Where can I find a definition?
1
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1answer
76 views

Braid invariants resource

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?
1
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0answers
39 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
23 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 ...
14
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1answer
188 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
33 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 ...
2
votes
1answer
39 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
votes
1answer
80 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
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0answers
40 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 $\mathbb{C^2}\cong \mathbb{R^4}$. ...
3
votes
1answer
75 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? ...
1
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0answers
114 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 ...
6
votes
1answer
327 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
49 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 ...
1
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1answer
174 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?
2
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1answer
73 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
49 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
33 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
47 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 ...
1
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1answer
121 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 ...
3
votes
1answer
50 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
208 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 ...
1
vote
1answer
196 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
472 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 ...
6
votes
2answers
174 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
votes
0answers
49 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
51 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
27 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
69 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
81 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
vote
0answers
103 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
vote
1answer
133 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
vote
1answer
78 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 ...
1
vote
1answer
122 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
302 views

Isotopy and Homotopy

What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.
1
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0answers
17 views

Skein trees resource

Is there a resource where a collection of Skein trees for the Conway polynomials of knots have been presented? Thanks
3
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0answers
65 views

Classify knots in a closed bead-spring like polymer simulation

my problem is to detect the crossing number (or another knot invariant) of a simulated polymer. A polymer is a closed bead spring, which mean that it is represented by a set of points connected by ...
7
votes
0answers
178 views

How did Chern pictured the first Chern number?

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
0
votes
1answer
46 views

Every cycle in a knot has odd length

Consider the projection of a knot on to the plane. Consider following the knot, starting from a crossing, until we get back to that crossing (on the opposite strand). Why must this cycle have odd ...
4
votes
1answer
59 views

Reidemeister III and minimal crossing knot

If you have a knot which has minimal crossings, can you do a Reidemeister III move? Thanks
2
votes
1answer
157 views

Dehn presentation proof reference request

Can someone give me a reference for a proof that the Dehn presentation of a knot group gives us the fundamental group of the knot complement in $S^{3}$?