# Tagged Questions

For questions on knot theory, the study of mathematical knots

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### 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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I get confused when checking the definition of Whitehead double. In this page, a Whitehead double is a special type of satellite knot. But how to understand the sentence "Take a Whitehead double of ...
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### 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!
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### Component-wise connected sum of links

Given two links $K = K_1 \cup \dotsb \cup K_n$ and $L = L_1 \cup \dotsb \cup L_n$, where each $K_i$ and $L_j$ are oriented knots, can we define the connected sum $K\#L$ by taking the connected sum of ...
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### Show that the cylinder is not ambient isotopic to the Mobius band.

Here is my definition for ambient isotopy: We say if there is an orientation preserving piecewise linear homeomorphism $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ (or replace $\mathbb{R}^3$ with $S^3$) ...
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### “Natural” Homeomorphisms, Retracts and Knots

I have been trying to prove the following result for a few days now, and have made some amount of progress, but now I'm struggling. This is what I'd like to prove: Every proper knot, K has a retract ...
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### 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. ...
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### Does knot theory provide any insight into the way real knots work?

Another user asked a well received but unanswered question: whether knot theory has lead to the development of better knots. His question is similar to mine in as much as both of our questions ask ...
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### Alexander-Conway polynomial of an unlinked knot…

I had asked this elsewhere earlier in the week but I decided I am more likely to get an answer here: Is it true that for all unlinks, the Alexander-Conway polynomial is equivalent to 0? It seems ...
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### 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 ...
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### On the definition of positive and negative crossings

In the definition of the Kauffman bracket, we have to resolve a crossing in two ways. The way to resolve these crossings involve distinguishing whether the crossings are positive or negative. But ...
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### 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 ...
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### 6-coloring of a knot

According to this page this knot should be 6-colorable (question 6): But I couldn't find an explicit coloring, which makes me think that the claim in the parantheses is not true. Can you find one? ...
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### 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 sail"-...
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### William Thurston's 'Knots to Narnia'

http://www.youtube.com/watch?v=IKSrBt2kFD4 Above is the youtube link to the short video called 'Knots to Narnia'. (9 mins) While learning knot theory, I found this interesting video. In the video, ...
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### Burau representation and the Jones polynomial

It is well known that if $K$ is the closure of a braid $f$ in the braid group $B_n$ then, up to some overall factor, the Alexander polynomial $\Delta_K(t)$ of $K$ is given by \...
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### Knots as boundaries

Boundary of a 2-manifold is a closed curve (or a set of closed curves), so I was thinking of reversing this process. In 2D space, a simple closed curve in a plane can be thought of as a boundary of ...
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### minimal diagram of a knot and the crossing changes

Given any knot diagam, we can deform it into unknot by switching over arc and under arc of some crossings in the diagram. My question is the following: Let K be a knot a diagram of a knot, suppose ...
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### Obtaining Wirtinger presentation using van Kampen theorem

Hatcher's Algebraic Topology, section 1.2, problem #22 describes an algorithm for computing the Wirtinger presentation of the complement of a smooth or piecewise linear knot K in $\mathbb{R}^3$: ...
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### Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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### Trapping a sphere in a trefoil knot

Is it possible to trap a sphere in the center of a trefoil knot? It seems like with three points of contact it should be possible. Admittedly not big into math, but a craftsman who loves trapping ...
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### Category theory for knot theory

Where would I start to learn category theory for its use in knot theory? I have a background in physics and Ive read Adams Knot book. I know nothing about category theory. Eventually I want to learn ...
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### Is there some knot theory behind the Mobius donut?

I was watching this video by Numberphile where a professor cuts a bagel into two interlocking pieces. Is this a torus knot or torus link? I'm trying to interpret in terms of $(p,q)$-torus knots Torus ...
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### How do we define the mirror image of a knot in general 3-manifolds

How do we define the mirror image of a knot in general oriented 3-manifolds ? For instance for a knot in an irreducible integer homology sphere.
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### Legendrian isotopy

I think the definition of Legendrian isotopy is that you can find an isotopy of the ambient manifold such that it takes a Legendrian knot to another through Legendrian knots. What I cannot figure out ...
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### fibered knots in $S^3$

Given a fibered knot $k$ in $S^3$, we have the decomposition of $S^3$ as union of $M$ and $S^1\times D^2$, where M is a fiber bundle over $S^1$, with fiber $F$ such that its boundary is the knot $k$. ...
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### Local Flatness and Knots

Peter Cromwell's `Knots and Links' has a definition for a locally flat knot that I'm struggling to understand. It's more the terminology used rather than the concept (I hope) but I can't find a ...
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### Is a knot shadow always compatible with the trivial knot?

Define a knot shadow as a projection of a knot that does not indicate over- and under-crossings. So, if there are $c$ crossings, there are $2^c$ possible over/under assignments, and so that many ...
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### Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
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### How to find the braids that when closed make the $6_1$ knot.

I have the $6_1$ knot and my question is how can I easily find the braids that when closed make this knot, what's the easiest way in general for any knot.
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### Correctness of the size of an planar integer lattice unknot

A planar integer lattice unknot is a polygon drawn over a two dimensional integer lattice. Here is an example: Given a number $N$, a planar unknot is not always possible. For example, a planar ...
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### 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 ...
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### Completeness of moves for polygonal knots

I am going through the paper, MINIMAL KNOTTING NUMBERS, by MANN et. al. On page six of the paper, they defined following moves for polygonal knots. Parallel moves Triangular moves I understand ...
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### 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 ...
Reading knot tables, it seems that as $n$ increases, more prime knots have crossing number $n$. Is this a proven fact? More precisely, If $k(n)$ is the number of knots with crossing number $n$, is ...
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, ~~~~~~~(\alpha,a)\...