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Questions tagged [knot-theory]

For questions on knot theory, the study of mathematical knots and their properties.

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Tubular neighborhood of knot

Is it always true that a tubular neighborhood of a knot $K \subset M$, where $M$ is a generic smooth 3-manifold, is diffeomorphic to $S^1 \times B^2$ (if you prefer diffeomorphic to $S^1 \times \...
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1answer
30 views

Question about Conway Polynomials of oriented links

Hi I've got a few questions on Conway polynomials in preparation for an exam this Saturday that I don't know how to do: Let $L$ be an oriented link. (a) If $\mu(L)=1$ then $C(L)\in 1+z^2\mathbb{Z}[z]...
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1answer
22 views

Question on linking number of weakly split links

Hi I don't have that many resources to learn this module and my exam is this Saturday. I'm having trouble proving the following: (a) Let $L$ be an oriented 2-component link of components $L_{1}, L_{2}...
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1answer
25 views

Do Reidemeister moves allow you to go from a prime knot of n crossings to another prime knot of n crossings?

I am trying to go through a set of Reidemeister moves that will go from knot $7(2)$ to knot $7(1)$. I am having trouble getting from one to another and was wondering if it is my lack of creativity or ...
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1answer
34 views

Open tubular neighborhood of $2$-disk in $4$-ball and its exterior

I try to be familiar with the notion of smoothly sliceness of knots and disks. A $2$-disk $D$ is said to be a slice disk if it a smoothly and properly embedded in $B^4$. The boundary of $D$ in $S^3$ ...
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1answer
21 views

Nomenclature for composite knots with hierarchies

I'm not a mathematician but highly interested in knots form a biological perspective, so I hope everyone is fine with me using a less mathematical formalism to express myself in this post! Recently, ...
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1answer
37 views

Implementing Creative Reidemeister Type II Moves

Question. Given a Gauss Code $C$ of a knot $K \subset S^3$ with respect to a diagram $D$, how can one determine all possible Reidemeister type II moves that increase the number of crossings in $D$? ...
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1answer
40 views

Does HOMFLY-PT polynomial for links ever vanish?

In my knot theory notes I have written down the following statement: Let $L$ be a link with $k$ components, then $P_L(x,y) - 1$ is divisible by $x+y-1$ and $P_L(x, y) + (-1)^k$ is divisible by $x+y+...
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17 views

Is the toroidal presentation of a grid diagram always the mirror image of the corresponding knot?

So I have tried changing the direction of some connecting lines in this grid representation of the right-hand trefoil knot: And then I identified the top edge with the bottom, and the left edge with ...
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1answer
62 views

Using grid diagram to compute the Alexander polynomial

I have been reading the book 'Grid Homology for Knots and Links' (see https://web.math.princeton.edu/~petero/GridHomologyBook.pdf) - in Section 3.3 it provided a way to compute the Alexander ...
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1answer
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Find a compact orientable surface whose boundary is the Hopf Link.

Consider the Hopf link given as $$ H_l = S^1 \times \{0\} \quad \cup \{ (x,0,z) \in \mathbb{R}^3 | (x-1)^2 + z^2 = 1\} $$ Find a compact orientable surface $\Sigma$ in $\mathbb{R}^3$ such that $\...
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1answer
262 views
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How to determinate the the number of crossing points?

This question is an extension of the question: how-to-determine-the-convergence-the-start-and-the-finish-points. One can apply the next algoritm and obtaine the ...
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0answers
23 views

When Alexander polynomial's span determines knot's genus?

Let $K$ be a knot with 3-genus $g$ and Alexander polynomial $\Delta_K(t)$. Define span of a polynomial in $t$ as difference between highest and lowest power of $t$, for example $$\operatorname{span} (...
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57 views

Borromean Weaving

Here's a picture by Rashmi Sunder-Raj. Is this topologically equivalent to Borromean Rings?
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1answer
56 views

Map of $\mathbb{R}^3-Knot \to S^1$

Reading Bachman's "A Geometric Approach to Differential Forms", in section 7.8.1 about the Lining Number invariant, I have stumbled upon the following assertion. Let the knot $K$ be defined as a (...
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1answer
37 views

Reference request for knot's signature via skein relation

Mathworld's article on knot signature [1] defines it as a function that satisfies two conditions: $s(K_+) - s(K_-) \in \{0, 2\}$ and $4 \mid s(K) \iff \nabla (K)(2i) > 0$. Where can I find proof ...
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1answer
62 views

“Human Knot” solvability probability

Somewhat surprisingly, I don't see a question about this. There is a team-building (or just fun mathematical) game where a group of people hold hands with each other, usually trying not to hold hands ...
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0answers
53 views

How is a wild knot an embedding of $S^1$?

All knot texts say the definition of knots as an embedding of $S^1$ to $R^3$ give the (mostly unwanted) posibility of wild knots, I'm wondering how so. The only way I can see it (and have found it) ...
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1answer
29 views

Reason for Z-axis orientation in torus knots

My understanding is that when |p|≠ 1 ≠|q| and they are coprime for (p,q) torus knots, those knots are chiral and while rotation and translation in three dimensions cannot map a chiral knot to its ...
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1answer
71 views

Why a special ball in $S^3$ is unique?

I'm studying : An introduction to knot theory(by: W.B. Raymond Lickorish). To prove composition of two oriented knots is unique, Lickorish has written:"regarded $K_1$ and $K_2$ as being in distinct ...
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1answer
38 views

Why does a a torus knot parameterization invert the sign of the z values?

The torus knot page on Wikipedia contains the following (p,q)-torus knot parametrization:\begin{aligned}x&=r\cos(p\phi )\\y&=r\sin(p\phi )\\z&=-\sin(q\phi )\end{aligned} where\begin{...
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Show Jones polynomial is unique

I am trying to show that if we define a new invariant of knots $W(L)$ which follows the same rules as the Jones polynomial $V(L)$, so that it has value 1 on the unknot and satisfies the Skein relation ...
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1answer
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The projection of a turning knot

If one imagines turning a knot, and looking at its projection on the plane, it will change between different projections of the same knot. In between, there will be some singularities when more then 2 ...
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Given a knot, what's the minimal genus of a torus the knot is embeddable on?

An n-embeddability definition appears towards the end of the section 5.1 Torus knots of the Knot book by C. C. Adams: A knot $K$ is an $n$-embeddable knot if $K$ can be placed on a genus $n$ ...
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1answer
41 views

Vogel's Algorithm - Why can we read braid words from nested coherent Seifert surfaces?

I was reading this paper from R. Goldstein-Rose: http://math.uchicago.edu/~may/REU2017/REUPapers/GoldsteinRose.pdf In Figure 12 it was mentioned that if a Seifert surface is coherent and nested, then ...
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1answer
43 views

torus crossing directions & Conway polynomial

I know that the Conway polynomial of a trefoil knot with all negative crossings is 1 + x^2. I was therefore wondering, would the Conway polynomial of an equivalent trefoil knot with all positive ...
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1answer
29 views

“The order of a torus link can be understood as a rational number”

The order of a torus link consists of a pair of integers $(m,n)$, with at least one of them nonzero, and it is such that if the two integers are not coprime, i.e. of the form $(km, kn)$, the link ...
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1answer
68 views

What is the braid word for the link L6n1

Please consider the link L6n1 Please note that such link is not the Borromean link L6a4. I am trying to obtain the braid word for L6n1. Using SnapPy with the following code ...
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2answers
213 views

Why can't I link these three loops together?

Today is Mardi Gras, so I decided to try to connect two of my friend's drawer handles together with a necklace. It is easy to connect the necklace to one drawer handle. (See picture for what I mean ...
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128 views

Is this link L10a169?

Please consider the following link I am using SnapPy with the following code ...
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Witten-Reshetikhin-Turaev (WRT) Invariant for surgery on the figure 8 at all roots of unity?

I've been trying to find a reference that gives the WRT invariant of a 3-manifold obtained by surgery on the figure 8 knot at an arbitrary root of unity but have only found them at the standard ...
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2answers
68 views

What link is this brunnian link?

Please consider the following brunnian link with four components I am trying to identify such link using SnapPy but SnapPy is not able to do it. My questions are : How many effective crosses such ...
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1answer
70 views

Fundamental group of torus knot without thickening

The calculation of the fundamental group of a $(m, n)$ torus knot $K$ is usually done using Seifert-Van Kampen theorem, splitting $\mathbb{R}^3\backslash K$ into a open solid torus (with fundamental ...
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37 views

How to prove that the mapping of the braid to the Automorphism group of the free group is a complete invariant?

Does anybody know how to prove that the mapping of the braid to the free group's automorphism group is a complete invariant?
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1answer
36 views

Prove that two variable Jones polynomial can be expressed by Finite type invariant

I have this question that says: Prove that two variable Jones polynomial can be expressed by Finite type invariant. Can somebody explain how this is done? many thanks in advance!
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51 views

Is it possible to cross a tangled rope?

So, I thought of this question a while ago and haven't found a clear answer for it yet. Here's the question: Suppose you have a rope that is tied to two ends of a room. You can tangle the rope ...
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1answer
61 views

Have similar theories like knot theory been developed in higher dimensions?

Well, my question is kind of basic but I hope it would be taken seriously by the community. Also, I'm very new to this topic and I want to study knot theory in future. Knot theory is the study of ...
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1answer
54 views

Equivalence of rational knots

I am trying to understand theorem 1.2 from http://homepages.math.uic.edu/~kauffman/VegasAMS.pdf: Theorem 1.2. Suppose that rational tangles with fractions $\frac{p}{q}$ and $\frac{p'}{q'}$ are ...
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2answers
72 views

Why is the fundamental group of the Hopf link abelian but a two component unlink isn't

Title says it all basically. I'm trying to understand why the fundamental group of the hopf link (or really, the compliment of the Hopf link) is abelian. I mean, in a certain way I understand it, but ...
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1answer
53 views

What singular link is this?

A knot is a smooth map $f:\Bbb S^1\to\Bbb R^3$. A link is a collection of knots which do not intersect, but are linked together. A singular knot is a smooth map $f:\Bbb S^1 \to \Bbb R^3$ whose image ...
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1answer
71 views

Extending $S^1$ to an embedding of $D^2$ in $\mathbb{R}^3$

Consider the specific embedding $f : S^1 \to \mathbb{R}^3$ given by, say, the unit circle in the $xy$-plane. Suppose further that this embedding is contained within a $3$-dimensional, simply-connected ...
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4answers
125 views

$\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ representation of $B_3$ braid group

I've been trying to find a representation of the braid group $B_3$ acting on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ but I can't find it anywhere. From what I understand I have to ...
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2answers
60 views

Number of minima in a ribbon disk?

I am asking this question mainly with the hope of finding a reference to (presumably well-trodden) topic. Let $K$ be a ribbon knot and define $I(K)$ to be the minimum over number of minima of all ...
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1answer
63 views

How to undo a type I Reidemeister move only using type II and type III moves?

I am pretty new to geometric topology and I am struggling to solve this problem. I have tried to approach this a lot of different ways and I will spare you the drawings. I am trying to undo this ...
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2answers
64 views

Definition of “transversal intersection” for piecewise linear submanifolds

I'm working with knots in the PL category. In "Surface Knots in 4-space" of Seiichi Kamada, the author states on p. 26 that the linking number of two oriented knots $K$ and $J$ in $S^3$ is the ...
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1answer
38 views

Are link (non-splittable) quandles complete invariant up to orientation?

I am reading knot quandles. I read that knot quandles are complete knot invariant upto orientation from thesis of David joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory, page ...
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31 views

Kauffman bracket for oriented links

When defining the Kauffman bracket, the skein relation is usually given for unoriented links. But if I put some orientation, one of the resolutions could potentially become contradictory. As an ...
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0answers
32 views

show that Jones two-variable polynomial can be expressed through finite type invariant. [duplicate]

I have this questions that says: prove that the two-variable Jones polynomial can be expressed through Finite type invariant. can somebody please explain how can this be done? thank you so much in ...
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1answer
49 views

Self-intersected curve and knot curve

I am new to knot theory. So the following problem description may not be rigorous; but I will try my best to explain intuitively. Please tell me if there is ambiguity. For any self-intersected curve $...
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1answer
33 views

Khovanov homology over vector spaces, Z-modules or groups?

When I read various papers on Khovanov homology, sometimes it is defined in terms of graded vector spaces, sometimes as graded $\mathbb{Z}$-modules. Is there a difference? E.g. can the vector field ...