For questions on knot theory, the study of mathematical knots

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Determine the multiplicity of knots for a graph

Here are my two questions: Given a finite connected non-oriented planar graph, is there a way to determine whether or not it is possible to derive a single non-trivial knot diagram from this graph, ...
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2answers
102 views

Recommended books on knot invariants

I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd ...
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1answer
114 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
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1answer
50 views

Req. for Definition:Twisting Number of Curve in Contact Structure

All: I'm reading a paper that makes mention of the twisting $tw (\gamma,S) $ , where $\gamma$ is a simple, closed Legendrian curve in a surface $S$ , and $S$ is embedded in a contact 3-manifold ...
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1answer
187 views

Why is the Whitehead double of a knot always prime?

I was looking for a proof that there are infinitely many prime knots and one said "take your favorite (prime) knot and consider all its Whitehead double", implying that all Whitehead doubles of a ...
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97 views

On the definition of Fox derivative

I am reading An Introduction to Knot Theory by W.B. Raymond Lickorish. In Chapter 11 the motivation for the Fox derivative is mentioned. I understand why the contribution of the occurrence of $x_j$ in ...
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0answers
355 views

Wirtinger Presentation for the Figure eight knot, Rolfsen exercise

I have been working through Rolfsen's "Knots and Links" and have found myself frustrated by exercise 4 on page 58. It concerns the Wirtinger Presentation of the figure eight knot, where the ...
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2answers
314 views

Total mean curvature of an immersed torus.

How to prove that the total mean curvature of an immersed torus of $R^3$ such that has nontrivial self-intersection must $> 8 \pi$? The definition of total mean curvature is the integral of $H^2$ ...
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1answer
501 views

Complement of figure-8 knot

I am reading W. Thurston's famous "3-dimensional Geometry and Topology", but I am stuck at the point where it is said that gluing two tetrahedra in an appropriate way give you the complement of the ...
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119 views

Zero exponent sum w.r.t group words in knot group's presentation

I am reading, "Plane Curves Associated to Character Varieties of 3-Manifolds" by Cooper, Culler, Gillet, Long, and Shalen and on page 28 ( http://www.math.uic.edu/~culler/papers/PlaneCurves/curves.pdf ...
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1answer
125 views

Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
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2answers
360 views

Uniqueness of Preferred Framing of a Solid Torus in $S^3$

One way to state my question tersely is: For a homeomorphism $f : S^1 \times \mathbb{D}^2 \rightarrow S^1 \times \mathbb{D}^2$, does $f|_{S^1 \times S^1}$ determine the isotopy class of $f$? This is ...
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1answer
67 views

Proof of the completeness of knot quandle

http://www.varf.ru/rudn/manturov/book.pdf I am reading p.56 in the book (p.69 in the pdf file), and trying to understand the proof that quandles completely determine knots up to orientation. The ...
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3answers
194 views

Software to calculate Alexander polynomials

Is there any software for Windows that I can use to calculate the Alexander polynomials of links?
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1answer
64 views

Knot group, Abelization and linking number

Suppose that $K$ is an orientable knot, $X=\mathbb{R}^3\setminus K$, $x_0\in X$ and $G=\pi_1(X,x_0)$. Suppose $\phi:G\rightarrow G_{ab}=G/G'\cong\mathbb{Z}$. Use the Wirtinger presentation of $G$ to ...
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2answers
178 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 ...
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1answer
44 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 ...
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1answer
250 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
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1answer
77 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?
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3answers
177 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 ...
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1answer
95 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} ...
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1answer
58 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 ...
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1answer
181 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 ...
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237 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 ...
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1answer
136 views

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

It seems to me, all three 3 bridges are needed?
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94 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 ...
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1answer
86 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 ...
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1answer
193 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) = ...
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0answers
79 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 ...
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3answers
409 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
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1answer
244 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 ...
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1answer
65 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 ...
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1answer
68 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
165 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 ...
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60 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 ...
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73 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 ...
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1answer
73 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 ...
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1answer
252 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 ...
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1answer
73 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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2answers
122 views

Burau matrix of braid

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

Braid invariants resource

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?
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44 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} ...
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0answers
28 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 ...
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1answer
196 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 ...
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1answer
35 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 ...
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1answer
41 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 ...
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1answer
103 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 ...
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44 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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1answer
93 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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122 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 ...