For questions on knot theory, the study of mathematical knots

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Conway polynomial of the unknot

I was trying to follow along with Wikipedia's basic computation of the Conway polynomial of the trefoil knot (http://en.wikipedia.org/wiki/Knot_theory#Knot_polynomials), but I got sidetracked by ...
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1answer
205 views

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ...
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96 views

Legendrian Isotopy of Knots can be extended to an ambient Contact Isotopy

I am attempting to understand a proof that an isotopy of two Legendrian knots $L_0$ and $L_1$ in a closed contact manifold (M,$\xi$) can be extended to an contact isotopy $\phi$ of M such that $\phi_0 ...
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81 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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55 views

What is a 2-surgery on a disk?

I am confused by a certain point in Scharlemann's paper "Sutured Manifolds and Generalized Thurston Norms", which seems important enough to not just skip it. I mean the "2-surgery on disks" in the ...
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17 views

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
92 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
96 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
48 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
149 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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81 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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301 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
281 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
436 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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110 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
96 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
345 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
58 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
157 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
57 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
160 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
39 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
175 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 ...
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1answer
67 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
156 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
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} ...
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1answer
53 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
139 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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193 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
89 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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91 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
84 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
155 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
74 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
321 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
190 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
61 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
64 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
146 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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54 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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0answers
68 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
71 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
211 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
70 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
104 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
79 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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41 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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26 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
192 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
34 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 ...