For questions on knot theory, the study of mathematical knots

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17
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0answers
184 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 ...
9
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0answers
130 views

Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
8
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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 ...
7
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0answers
50 views

Is “slightly deform” a well defined concept in mathematical proof?

In topological proofs the phrase "slightly deform" is widely used. To me, although I can accept the idea intuitively, the phrase "slightly deform" does not sound like a strict mathematical concept. ...
5
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310 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 ...
4
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31 views

Can information about a knot be recovered from the Jones Polynomial?

Suppose we know the Jones polynomial of some knot, but maybe not specifically which knot. Can any information about the knot be recovered just by knowing its Jones polynomial? Say, for example, the ...
4
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0answers
46 views

Has knot theory led to the development of better knots?

Knot theory was likely originally motivated by the study of real-world knots such as these: Indeed, mathematical knot tables to this day look not too dissimilar from the familiar "age of ...
4
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0answers
70 views

Finding the jones polynomial of the following knot

I am struggling with finding the Jones polynomial of the following knot using only the sklein relations I am also asked if it is Isotopic to its mirror image. I tried using the above relation, ...
4
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0answers
188 views

Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
4
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0answers
43 views

Name of a link invariant?

Below I will describe a link invariant, denoted by me as $inv(L)$. Has anyone encountered this invariant in the literature? If so, what is its name? Also, any references to papers or books that ...
4
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0answers
54 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 ...
4
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0answers
105 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 ...
4
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0answers
98 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(
3
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0answers
32 views

third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$

It is known that the third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$. Each colouring $C$ of a knot diagram by a dihederal quandle $R_3$ induces a homomorphism $f:C \rightarrow ...
3
votes
0answers
56 views

Actual Definition of the term: Hopf Band?

Sorry if this is too trivial: I need an actual working definition of the term: Hopf band. I see references to it in many searches, but never an actual precise definition. All I know so far is that ...
3
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0answers
81 views

Crossing bound implies Reidemeister move bound?

In 1998 Galatalo established an upper bound on the number of Reidemeister moves needed to convert a diagram $D$ of the unknot into a trivial loop diagram. The upper bound is a function of $n$, the ...
3
votes
0answers
114 views

Why can't I tie a infinite rope in hard knots?

I think this is a genuine math problem. And it's somehow related to knot energy but not directly solved by the latter. Why can't I tie a hard knot on a rope of infinite length? By infinity I mean ...
3
votes
0answers
79 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 ...
3
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0answers
77 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 ...
3
votes
0answers
85 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 ...
3
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0answers
52 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 ...
3
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0answers
40 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 $\mathbb{C^2}\cong \mathbb{R^4}$. ...
3
votes
0answers
50 views

does a method exist to distinguish two component link consisting of just two unknots from an unlink?

Clearly, linking number is not enough as there are links like whitehead. There is the enhanced linking number based on conway polynomial that can distinguish whitehead (and infinite family of such ...
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
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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 ...
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
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, ...
3
votes
0answers
77 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
0answers
399 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
0answers
54 views

Kauffman Bracket

Can anyone explain or direct me to: . What are the x and y's? Is x cutting the loop? And for the last picture, I don't understand how the Reidemeister moves have been applied there. I would much ...
2
votes
0answers
62 views

How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
2
votes
0answers
79 views

Why is surgery along a framed link well defined?

Let $L=L_1 \cup L_2 \cup \cdots \cup L_n $ be a framed link in $ S^3 $. I want to perform the surgery along $L$ to get a new manifold $M$. By definition, to perform this surgery, I must perform the ...
2
votes
0answers
45 views

Cohomology-Homology bilinear form of Seifert surfaces

Let $C_\ast$ be any chain complex of $R$-modules. Then for any $k\in\mathbb{Z}$ we obtain a $R$-bilinear map $$\langle-,-\rangle:H^k\!C_\ast\times H_kC_\ast\longrightarrow R, ...
2
votes
0answers
52 views

Seifert surfaces in Riemannian manifolds?

Does there exist an equivalent to Seifert surfaces for other Riemannian manifolds than $\mathbb{R}^3$? More precisely: Let $M$ be a simply-connected Riemannian manifolds and $K \subset M$ a (tame) ...
2
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0answers
29 views

diagrams of twist spun torus knots

Kindly can you explain to me how to obtain the double twist spun of torus knots from tangle diagram of the given torus knot. I found the method here ...
2
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0answers
58 views

Double coloring to distinguish mirror images

Recently, there was an interesting blog about distinguishing the right-handed trefoil from the left-handed trefoil using a variant of tricolorability (found in the following link): ...
2
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0answers
37 views

Piecewise linear knots and smooth knots

Is the set of all piecewise linear (PL) knots is a good approximation of the set of all 1D smooth knots embedded in $\mathbb{R}^3$? Once I saw a theorem related to that but not able to find it now. ...
2
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0answers
38 views

Knowing the existence of a fixed point set from an induced fundamental group automorphism

Let $L$ be a link in $S^{3} $ and $f_{ \phi } : \pi_{1} (S^{3} \backslash L ) \rightarrow \pi_{1} (S^{3} \backslash L )$ be induced from a periodic map $\phi $ of $S^{3} $, restricted to the ...
2
votes
0answers
74 views

How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
2
votes
0answers
90 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 ...
2
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0answers
291 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 ...
2
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0answers
109 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 ...
2
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0answers
66 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 ...
2
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0answers
38 views

Seifert surface and crossing number

i am sitting here with the problem of Seifert Surfaces. I know from a theorem that every knot does have a Seifert surface. We can also make a so called disc-and-band surface $F$ by gluing $v$ discs ...
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
71 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 ...
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. ...
2
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0answers
105 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 $C(p_{1},...,p_{n})$ ?
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0answers
32 views

Knots which are composed of several strands

In a math textbook and this article in NRICH, some problems deal with a special kind of knots: those which are formed from several strands: The problems ask if a given knot can be formed from just ...
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0answers
19 views

What is the connection (if any) between Knot Theory and Fluid Dynamics?

I've heard there is a connection to physics, but I'm unsure about any specific connection to fluid dynamics. Thanks!