For questions on knot theory, the study of mathematical knots

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Is there a Knot Theory software to analyze general curves in 3D?

So I happen to like proteins quite a lot and one thing that is very similar to a protein, when represented as the bare minimum, is a 1D curve embedded in the 3D space. They form beautiful and unique ...
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1answer
40 views

Jones Polynomial from Statistical Mechanics

I've been told that, given a knot projection, there is a way of associating a statistical system in such a way that the partition function of the system corresponds to the Jones polynomial of the ...
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1answer
39 views

Are Knots closed?

In every definition I see, a (classical) knot is an embedding of $S^1$ in $S^3$ or $\mathbb{R}^3$. But my lecturer said that the complement of a knot in $S^3$ is open, hence the knot is closed. But ...
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1answer
31 views

Equivalence of unoriented knots by ambient isotopy

I'm trying to understand the equivalence of unoriented knots in oriented 3-manifolds for my thesis, and getting confused. I have not found a satisfactory definition of this equivalence. My ...
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2answers
82 views

Which mathematical topics is knot theory related to?

I wonder if knot theory is related to any other topic in mathematics. I've not read much about it, but it seems to be living isolated. I also wonder if there any particular mathematical background ...
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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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1answer
30 views

Whitehead double

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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37 views

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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27 views

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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1answer
28 views

“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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Jones Polynomial of the Torus Knot $(2,n)$

I am working to derive the formula for the Jones polynomial of the torus knot (or link) $(2,n)$. A $(p,q)$-torus knot is obtained by looping a string through the hole of a torus $p$ times with $q$ ...
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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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2answers
42 views

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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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 ...
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1answer
36 views

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 ...
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1answer
82 views

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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1answer
34 views

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 \begin{equation} ...
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1answer
30 views

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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2answers
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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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1answer
50 views

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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66 views

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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smoothing a triple point by 1-handle addition

Given a surface diagram of a surface knot, we can attach 1-handle between the bottom and the middle sheet or the top and the middle sheets, in both cases a triple point is smoothed. My question is ...
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130 views

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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2answers
81 views

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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1answer
78 views

Probability that two circles in space are linked

Let $C_0$ be a circle centered on the origin, and $C_1$ a circle centered on $(1,0,0)$, center distance of $1$. Q1. If both $C_0$ and $C_1$ are randomly oriented and have the same radius $r ...
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1answer
27 views

Bridge index of Pretzel link

I was studying the bridge number of various kinds of links.I have heard there is some correlation between the bridge index of a Pretzel link and it's representation.Can anyone please explain it or ...
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2answers
65 views

is knot type invariant under diffeomorphism?

Is it possible to have a diffeomorphism of $R^3$ which changes the knot type, for instance the image of a trivial knot is a trefoil knot?
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1answer
100 views

Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
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1answer
54 views

How do we know how many prime knots there are for a specific number of crossings?

For instance, on wikipedia it says there are 7 prime knots with 7 crossings. How do we know there isn't an 8th prime knot knot with 7 crossings that we haven't yet discovered?
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A comment on a proof of equivalent knots

The following theorem and its proof is from A First Course in Algebraic Topology By Czes Kosniowski pp. 219-220 ...
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1answer
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Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
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1answer
81 views

How many ways are there of tying a tie?

I am sorry if this is useless. I have read in a newspaper that mathematicians have found the number of ways a tie can be tied. How could such a problem be solved? I'm asking out of curiosity.
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1answer
87 views

Why a torus knot is a prime knot?

Why a torus knot is a prime knot?
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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, ...
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1answer
174 views

two interlocked circles are homeomorphic to two noninterlocked circles

This is what I learned from here the post: two interlocked circles are homeomorphic to two noninterlocked circles, thus they (two interlocked circles and two noninterlocked circles) are homotopic ...
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1answer
69 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 ...
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148 views

Homology of knot complement

I was told in a topology class that if $Y$ is a closed $3$-manifold and $K$ is a null-homologous knot in $Y$, then $H_1(Y- \nu(K)) \cong H_1(Y) \oplus \mathbb{Z}$. I'm trying to prove this ...
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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 ...
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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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1answer
21 views

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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1answer
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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 ...