For questions on knot theory, the study of mathematical knots

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4
votes
1answer
216 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
160 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
357 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
101 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
295 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
392 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
117 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
417 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
248 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
173 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
4answers
799 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
270 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
433 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
113 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
262 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
314 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
233 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
115 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
272 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
102 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
75 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
120 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
245 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
339 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
185 views

Weird Diagram in Knot Theory

Can anyone make sense of fig 4.8? $\hskip 0.5in$ (Google books link.)
0
votes
1answer
146 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
368 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
160 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
143 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
144 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
93 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
414 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
360 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 ...
6
votes
2answers
181 views

The construction of knotted surfaces in $\mathbb{R}^4$

For a two-sphere embedded in $\mathbb{R}^4$,how can you check whether or not there is an ambient isotopy to the "standard" 2-sphere (the set of points $(x,y,z,0)$ in $\mathbb{R}^4$ distance 1 from the ...
3
votes
3answers
597 views

School project in knot theory

Can someone suggest an idea for a school project in knot-theory for a 13 year old? Thanks
10
votes
8answers
957 views

A good quick introduction to Knot Theory?

Is there a good quick introduction to knot theory? I am relatively mathematically savvy so any level is appreciated.