For questions on knot theory, the study of mathematical knots

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5
votes
2answers
159 views

Framed Cobordism Classes of links in $\mathbb R^3$

We know that every link in $S^3$ is framed cobordant to the unknot with some framing. The idea is to study smooth homotopy classes of maps from $S^3$ to $S^2$. Actually in the title I have given ...
1
vote
1answer
38 views

Isotopy of links

If a link $L$ has $\mu$ components $K_1,\ K_2,\ ... K_\mu$ and $L'$ has components $K'_1,\ K'_2,\ ... K'_\mu$ components, does "$L$ is isotopic to $L'$ " imply that "each $K_i$ is isotopic to ...
3
votes
1answer
157 views

trefoil knot and meridian/longitudinal cycles

I hope this is a simple question... For the trefoil knot 3_1, whose knot group is given by a presentation of the fundamental group, $\pi_1(M) = \langle a,b: aba = bab \rangle$, where the meridian and ...
4
votes
1answer
65 views

Computing the volume of a hyperbolic knot

Could anyone show me or refer me to a link where the volume of a hyperbolic knot, say, the figure-8 knot, is computed (well, in fact estimated) explicitly and not only having the procedures outlined?
1
vote
3answers
144 views

The Abelianization of $\langle x, a \mid a^2x=xa\rangle$

I wish to verify the following statement (which comes from Fox, "A Quick Trip Through Knot Theory", although that is probably not important). "$\Gamma=\pi_1 (M)=\langle x, a \mid a^2x=xa\rangle$ so ...
2
votes
1answer
85 views

Gaussian linking coefficient definition

As I read from the wikipedia page of the linking number, it says that the linking number of two curves $\gamma_1$ and $\gamma_2$ in space can be found using the integral $$\,\frac{1}{4\pi} ...
2
votes
1answer
52 views

Question on linear maps defined in Khovanov homology

There are two linear maps $m:V \otimes V \rightarrow V$ and $\Delta:V \rightarrow V\otimes V$ in the definition of the differential of Khovanov homology. So my question is why do they map elements as ...
4
votes
1answer
137 views

Finding the Alexander polynomial of the following braid closure

How do you find the Alexander polynomial of the closure of the following braid, $(\sigma_1^{-2}\sigma_2^{-1}\sigma_3^{-1}\sigma_4^{-1}\sigma_5^{-1}...\sigma_{A-1}^{-1})^B$ where $A$ and $B$ are ...
17
votes
0answers
184 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
2
votes
1answer
87 views

Why is the bridge index of the trefoil equal to 2?

It seems to me, all three 3 bridges are needed?
3
votes
0answers
86 views

Prove that $S^{1}$ unknots in $\mathbb{R}^{4}$

The definition is, X unknots in Y if any two embeddings are equivalent. How do you show $S^{1}$ unknots in $\mathbb{R}^{4}$ and in general, $S^{n}$ unknots or knots in $\mathbb{R}^{m}$? The solution ...
2
votes
1answer
83 views

Density of continuous knots in the plane transversal to some circles

This is an exercise from the book "Knots and Links" by Rolfsen (exercise 6 in section 2C) Let $\kappa : S^1 \rightarrow \mathbb{R}^2-(0,0)$ be a continuous imbedding. Let $M := \{ x \in ...
2
votes
1answer
146 views

On the HOMFLY polynomial of a split link

Basically my question has to do with the HOMFLY polynomial. In its wikipedia page, http://en.wikipedia.org/wiki/HOMFLY_polynomial, I see that it says $$P(L_1 \cup L_2) = ...
1
vote
0answers
70 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 ...
9
votes
3answers
304 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
votes
1answer
180 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
vote
1answer
60 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
62 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
145 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
52 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
votes
0answers
66 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 ...
4
votes
1answer
200 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
votes
1answer
69 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
102 views

Burau matrix of braid

What is the definition of a Burau matrix of a braid? Where can I find a definition?
1
vote
1answer
78 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
vote
0answers
40 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
vote
0answers
25 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
votes
1answer
191 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
86 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
41 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
80 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
vote
0answers
117 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
406 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
vote
0answers
54 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
vote
1answer
193 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
votes
1answer
75 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
vote
0answers
50 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
38 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
vote
1answer
48 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
vote
1answer
137 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
213 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
210 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
494 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
185 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
50 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 ...