For questions on knot theory, the study of mathematical knots

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3
votes
1answer
146 views

Knots in $S^1\times S^2$

Is there any special study of knots in this particular 3-manifold? A more targeted / simple question: What are some nontrivial examples of knots $S^1\subset S^1\times S^2$, and is there convenient ...
1
vote
1answer
110 views

The definition of a knot

The definition of a knot is an injective piecewise linear map from $S^{1} $ to $\mathbb{R}^{3} $. Isn't that equivalent to a subset of $\mathbb{R}^{3} $ homeomorphic to $S^{1} $ that is piecewise ...
2
votes
1answer
71 views

Computational complexity of unknotting problem?

The Wikipedia article on the unknotting problem says "a major unresolved challenge is to determine [...] whether the problem lies in the complexity class P". It mentions some work towards this result ...
1
vote
1answer
67 views

Virtual knot diagrams on surfaces with genus?

To the best of my limited understanding, a virtual knot diagram may be thought of as the projection of an embedding of $\mathbb{S}^1$ in a 2-manifold with genus onto $\mathbb{R}^2$. That is to say it ...
3
votes
0answers
82 views

Can the HOMFLY polynomial be obtained from the Kauffman Polynomial for torus knots?

This is essentially a yes/no/reference request question. Let me first just ask my question: Is there a known relationship between the HOMFLY and Kauffman polynomials of torus knots? In particular, ...
2
votes
0answers
81 views

Proof of the existence of a special parametrization of the trefoil knot.

Could someone kindly exhibit a periodic $ C^{2} $-function $ f: \mathbb{R} \rightarrow \mathbb{R} $ with period $ 1 $ such that $$ \{ (f(x),f'(x),f''(x)) \,|\, x \in [0,1] \} $$ represents a trefoil ...
2
votes
0answers
67 views

L(q,p)# L(p,q) = surgery on pq-torus knot complement

I am trying to solve one of Rolfsen exercice. That is prove that the connected sum $L(q,p)\# L(p,q)$ can be obtain by surgery on the complement of the pq-torus knot in $S^3$. I am doing it using ...
6
votes
2answers
118 views

How does smoothness prevent “singularities”?

This is a refinement of one of my earlier questions (I failed to put into words what I really wanted to ask). First of all, I'm not sure "singularity" is the correct word to use hence the quotes. ...
4
votes
2answers
171 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
4
votes
1answer
220 views

Question about definition of regular projection of a knot

One can define a knot in two ways: (1) A knot is a closed polygonal curve in $\mathbb R^3$ (2) A knot is an equivalence class of embeddings $S^1 \hookrightarrow \mathbb R^3$ And perhaps also: (3) ...
4
votes
1answer
163 views

Assigning alternate crossings to closed curves

This is a minor curiosity that I've been wondering about. Suppose that we draw a closed curve in the plane and that this curve intersects itself several times, but never twice in one spot. We can knot ...
0
votes
1answer
65 views

Follow up on cinquefoil knot

Using the following Seifert surface of the cinquefoil knot I get the following Seifert matrix (of linking numbers): $$ S = \begin{pmatrix}- 1 &1 &0 &0 \\ 0 &-1 &1& 0 \\ ...
3
votes
1answer
369 views

Framed manifolds and framed knots

I've been looking at intersection forms and the Arf invariant recently and I got a comment to one of my previous questions related to this. So I looked at framed manifolds. There seems to be quite a ...
1
vote
1answer
103 views

Showing pass equivalence of cinquefoil knot

According to C.C. Adams, The knot book, pp 224, "every knot is either pass equivalent to the trefoil knot or the unknot". A pass move is the following: Can someone show me how to show that the ...
7
votes
2answers
299 views

Seifert matrices and Arf invariant — Cinquefoil knot

I have computed the following Seifert matrix for the Cinquefoil knot: $$ S = \begin{pmatrix} 1 & -1 & -1 & -1 \\ 0 & 1 & 0 & 0 \\  0 & 1 & 1 & 0 \\ 0 ...
7
votes
2answers
415 views

Seifert matrices — Figure 8 knot

I've just learnt about Seifert matrices and thought it might be a good idea to compute some. Can you tell me if this is right: Here $x_1^+$ denotes the push off of $x_1$. I have omitted the diagram ...
2
votes
1answer
121 views

Embedded circles in $n$-dimensional space

A knot can be defined as an embedded circle in $3$-dimensional Euclidean space or in the $3$-sphere $S^3$. There is also a notion of a knot in higher dimensions: an $n$-knot is an embedding of the ...
4
votes
1answer
441 views

Braid groups and the fundamental group of the configuration space of $n$ points

I am giving a lecture on Braid Groups this month at a seminar and I am confused about how to understand the fundamental group of the configuration space of $n$ points, so I will define some ...
4
votes
2answers
252 views

Self-Linking Number on 3-Manifolds

We can assign a framing to a knot $K$ (in some nice enough space $M$) in order to calculate the self-linking number $lk(K,K)$. But of course it is not necessarily canonical, as added twists in your ...
0
votes
1answer
105 views

Pure braid group, stabilizer

From group theory we know that a homomorphism $\phi: G \to \operatorname{Sym}(S)$, where S is a set, then $\operatorname{Sym}(S) \cong \Sigma_n $. Its kernel is given as $\bigcap_{s \in S}G_s$, which ...
2
votes
0answers
178 views

Tying knot theory with traveling salesman problem (TSP)

If you draw a knot and place lots of evenly-spaced points on it, with straight segments between adjacent points, clearly the knot you started with is the shortest solution to the TSP in 3 dimensions. ...
12
votes
5answers
869 views

Trefoil knot as an algebraic curve

Is the trefoil knot with its usual embedding into affine $3$-space an algebraic curve (maybe after extending scalars to $\mathbb{C}$)? Is there even some thickening to some algebraic surface? If ...
5
votes
0answers
286 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
11
votes
1answer
470 views

Why is the knot group of the trefoil isomorphic to the group of 3-braids?

I apologise in advance for the vagueness of this question but I have not been able to find very much info on the topic and have made very little progress on my own. I am trying to understand why the ...
2
votes
1answer
80 views

Remove links by Kirby moves

I am trying to prove the following proposition. proposition; If in a framed link $L$ a component $K$ is an unknot with framing zero which links only one other component $H$ geometrically once, ...
2
votes
1answer
116 views

How to detect a twist or framing in a 3-manifold.

This question is somewhat a continuation of the question Gluing a solid torus to a solid torus with annulus inside. If we consider a genus one handlebody $U$ with an (nontwisted)annulus inside the ...
1
vote
0answers
88 views

Embedding of $T^{2}$ in $S^{1}\times S^{2}$.

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map and $i_{*}:\pi(T^{2})\rightarrow\pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times ...
7
votes
4answers
266 views

Reference for an unknotting move

Consider the following move on diagrams. I dimly recall hearing or reading that a sequence of such moves is sufficient to unknot any knot but I don't recall where I saw this. The strands in the ...
10
votes
1answer
317 views

Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question has been cross posted on MathOverflow with some very interesting answers and discussion. I'm currently writing a project on the braid groups and their analogues on closed surfaces. ...
4
votes
1answer
239 views

Computing knot/link groups

The knot group of a knot $K$ is the fundamental group of $\mathbb R^3 \smallsetminus K$; that is, the set of possibly self-crossing closed paths (starting and ending at any single point in space) ...
4
votes
1answer
117 views

What knot groups are Abelian?

The knot group (the fundamental group of the complement of a knot) of the unknot is $\mathbb{Z}$ and the Hopf link is $\mathbb{Z}^2$, so those are knots (links) with Abelian knot group but are there ...
3
votes
1answer
277 views

Surgery, framing and Dehn twist

Let $L$ be a framed knot in $S^3$. Let $U$ be a closed regular neighborhood of $L$ in $S^3$. How can I interpretate the following sentence? "We identify $U$ with $S^1 \times B^2$ so that $L$ is ...
1
vote
1answer
122 views

Identifying Prime Knots

Given two knots $K$ and $L$. With Seifert matrices $M_{K}$ and $M_{L}$ respectively, then the matrix $\begin{bmatrix}M_K & \\ &M_L\end{bmatrix}$ is a Seifert matrix of the connected sum ...
3
votes
0answers
103 views

Genus of fattened knots and links

Imagine you take a knot or link, and "fatten" it as much as possible, so that the surface is snug against itself at several places. Then "fuse" it into a solid object. What I have in mind is ...
3
votes
0answers
76 views

If a strip of paper is knotted into an open trefoil, what is the linking number?

It is assumed that the paper strip is knotted into an open trefoil (forming a pentangle) that lies flat on the table, and that the two ends of the paper strip are continued up to spatial infinity. ...
3
votes
1answer
123 views

Number of knots possible with length L string

What is the asymptotic growth in L for the numer of topological different knots possible using a length L closed string of radius 1? In 3 dimension euclidean space.
3
votes
1answer
258 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
2
votes
0answers
104 views

Rational Links and Conway Normal Form

Is it true that a link is rational if and only if it has a diagram of the form $C(p_{1},...,p_{n})$ ?
3
votes
2answers
351 views

Graphical-entry knot theory program for Mac?

Is there a good program that runs on Mac OSX, which has a graphical interface for inputting knot or link diagrams, and calculates standard invariants like the Conway and Jones polynomials? I have ...
1
vote
0answers
188 views

Weird Diagram in Knot Theory

Can anyone make sense of fig 4.8? $\hskip 0.5in$ (Google books link.)
0
votes
1answer
148 views

Calculating the probability that a cord makes a knot

I use earphones to listen to music. The cord connected to the earphones often gets entangled in my pocket and makes a knot, which I always find hard to untangle. How can I define and calculate the ...
3
votes
0answers
376 views

Ambient Isotopy

From Hirsch's Differential Topology, p. 180. The first of the isotopy extension theorems says; Let $A\subset M$ be a compact submanifold and $F:V\times I \rightarrow S^{3}$ an isotopy of $A$. If ...
2
votes
1answer
175 views

Locally flat submanifold

Recently I found the following definition: Let $M^{n}$ be an $n-$dimensional topological manifold. Then $N^{k}\subseteq M^{n}$ is a locally flat submanifold if for every $x\in N$ there exists an open ...
4
votes
1answer
145 views

knot invariants from the symmetric group

Producing knot/link invariants is "as simple as" finding functions on the braid group invariant under the Markov moves. Many classical invariants arise from the character of a representation of the ...
1
vote
1answer
150 views

Are satellite knots prime?

Which satellite knots are prime? I do know that connected sum of knots is a satellite operation, but I found this statement: "the satellite knots all have structures which are well known and ...
4
votes
0answers
94 views

Knots with Fox tricoloring number tri(K)=27

I would be very grateful if you help me to find such knots. Or to find a knot atlas, where this invariant is included. I tried to find, but did not sucseed(
0
votes
1answer
67 views

Positive genus and Abelian-ness

So if a Seifert surface of a knot K has positive genus, what can we say about the its complement's first fundamental group, is it Abelian because the only knot with Seifert surface of positive knot is ...
0
votes
3answers
426 views

restrictions on the fundamental group of a knot complement?

Are there any restrictions on $\pi_1(S^3\backslash K)$ for a (tame) knot $K$ besides having $\pi_1^{\text{ab}}(S^3\backslash K)=\mathbb{Z}$? So we have 1) finite presentation, 2) $H_1=\mathbb{Z}$, 3) ...
4
votes
1answer
332 views

Pretzel knot equivalence

How would you go about proving the $(p, q, r)$ -pretzel knot is equivalent to the $(p, r, q)$ -pretzel knot? By "equivalent" I mean you can change one knot into the other by elementary deformations. ...
12
votes
2answers
364 views

Why are knot invariants best organized as polynomials?

Does anyone have a good explanation for why Knot invariants tend to be well organized as polynomials? What exactly is going on and why don't we often see polynomial invariants for classifying other ...