For questions on knot theory, the study of mathematical knots

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3
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0answers
29 views

$3\tau(K_1$#$K_2)$=$\tau(K_1)\tau(K_2)$

Suppose we have two knots $K_1$ and $K_2$. Then look to the connected sum of $K_1$ and $K_2$ denoted by $K_1$#$K_2$ (defined for knots). Suppose $\tau$ is the number of $3$-colourings (definition for ...
3
votes
3answers
70 views

Differential characterization of unknots

How can the closed simple curves in $\mathbb{R}^3$ be characterized that can be boundaries of a 2-dimensional oriented surface in $\mathbb{R}^3$? Intuitively I would tend to say that it's exactly the ...
1
vote
0answers
82 views

Linking number and the factor $\frac{1}{2}$

i have a question about the linking number of a knot. Per definition: The Total Linking Number Lk(D) is obtained by taking half sum over all crossings (for more definition look to other definitions ...
1
vote
0answers
113 views

Torus link and knots

Hello :) i am reading about knot theory especially torus links :) i read "Crossing number and Torus links" and the answer isn't clear. Does there exist a solution without topology but with group ...
1
vote
1answer
138 views

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
1
vote
1answer
83 views

Why is there no contradiction by construction of alternating knots? [duplicate]

I have got a question. From definition alternating diagram $D$ of a knot $K$ is a diagram such passes alternately over and under crossings. A knot $K$ with such a diagram $D$ is called a alternating ...
1
vote
1answer
131 views

Crossing number and Torus links

We define the crossing number of a knot $K$ to be the minimal number of crossings in any diagram of $K$. Surely we can easy prove that there do not exist knots with crossing number $1$ and $2$ ...
2
votes
1answer
379 views

Isotopy and Homotopy

What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.
1
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0answers
17 views

Skein trees resource

Is there a resource where a collection of Skein trees for the Conway polynomials of knots have been presented? Thanks
3
votes
0answers
67 views

Classify knots in a closed bead-spring like polymer simulation

my problem is to detect the crossing number (or another knot invariant) of a simulated polymer. A polymer is a closed bead spring, which mean that it is represented by a set of points connected by ...
8
votes
0answers
182 views

How did Chern pictured the first Chern number?

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
0
votes
1answer
46 views

Every cycle in a knot has odd length

Consider the projection of a knot on to the plane. Consider following the knot, starting from a crossing, until we get back to that crossing (on the opposite strand). Why must this cycle have odd ...
4
votes
1answer
62 views

Reidemeister III and minimal crossing knot

If you have a knot which has minimal crossings, can you do a Reidemeister III move? Thanks
2
votes
1answer
171 views

Dehn presentation proof reference request

Can someone give me a reference for a proof that the Dehn presentation of a knot group gives us the fundamental group of the knot complement in $S^{3}$?
1
vote
0answers
55 views

Proof of uniqueness of decomposition into prime knots

I would like to know the proof of uniqueness of decomposition into prime knots is for a given knot or for a given equivalence class of knots and how to see it. Thanks.
1
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0answers
30 views

Knot program crossing information

Is there a program where if I just give information on a knot-things like the crossing information, then the program will produce the knot?
3
votes
0answers
44 views

Linking integral unchanged over continuous deformations

Say we first have two curves, $C_1$ and $C_2$ which are knotted together. Let $C_2'$ be a continuous deformation of $C_2$ such that $C_2$ does not cross $C_1$ as it is deformed into $C_2'$. How ...
3
votes
1answer
149 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 ...
2
votes
1answer
118 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
74 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
73 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
89 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
72 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
121 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
175 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
245 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
169 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
67 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
408 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
108 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
315 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
450 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
127 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 ...
6
votes
1answer
490 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
288 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
109 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
180 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. ...
13
votes
5answers
981 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
311 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
539 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
81 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
120 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
89 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
276 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
323 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
257 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
122 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
290 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
127 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 ...