For questions on knot theory, the study of mathematical knots

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23 views

How do you show that the 6 3 knot is a reversible knot? [closed]

Or any knot in general. Without drawing it out to show reversibility
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15 views

Smoothing multiple crossings

Can you smooth multiple crossings at once in a knot when computing the bracket polynomial? If so is a there a rule for this?
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1answer
21 views

Bracket polynomial of a hopflink

When you are computing the bracket polynomial of the (for example) hopflink, why can you not smooth all the crossings in one go? Why do you have to only first start with one crossing? For example in ...
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0answers
9 views

Linking number and writhe of the hopflink

If you have two circles linked up to give the hopflink. Both these circles have clockwise orientation. Is the linking number and writhe going to be zero?
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0answers
11 views

Reidemeister infinity move

If you have a knot K. How do you apply a R infinity move to it? Is their an algorithm which tells you what to do at certain parts on a knot?
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4answers
271 views

An Illustrated Classification of Knots.

Let me be honest here: I know very little about Knot Theory. I'm sorry. I've a friend though, someone with no training in Mathematics at all but who is a huge fan of knots (for whatever reason), who ...
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1answer
27 views

embedding projective plane in 4-space? [closed]

Is it possible to embed projective plane in 4-space? If not what is the reason and what is the smallest singularity set?
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16 views

Compatible seifert circles

When are two seifert circles compatible? I originally thought that it was just a matter of looking at their orientations to check but it seems that is not enough.
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0answers
26 views

Is there a complete Link Invariant for links with N crossing.

Are there known examples of pairs $\left(f, N\right)$, where $f$ is a link invariant that is known to be complete when restricted to link diagrams that have at most $N$ crossings? (Ideally, f should ...
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1answer
51 views

math in horseshoe puzzle

We know that Rubik's Cube is a good demonstration of group theory. Correspondingly, for the horseshoe puzzle as in the picture below, is there a math language for it? Does it demonstrate any math ...
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39 views

Is it trivial that you cannot get apart 2 linked unknots?

I have read that: a trefoil is not an unknot is not trivial in knot theory. And it put the following question in me. Is it also not trivial that you cannot get apart 2 linked unknots? I mean, is it ...
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2answers
32 views

Knots isotopic to their mirror image

How do you prove a knot is achiral? Do you just show swap the under crossings and the over crossings?
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1answer
75 views

Is there any knot showing infinitely many crossings?

Let $K$ be any unknot. Is it possible that $K$ shows infinitely many crossings? And if it is possible: How to get $K$ from the simplest unknot through Reidemeister moves?
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1answer
40 views

Are there knots that can be distinguished between by the Jones polynomial that the HOMFLY polynomial cannot distiguish?

Obviously the HOMFLY polynomial is a more superior invariant than the Jones polynomial since it is a geralisation of the Jones. However, are there knots that can be distinguished between by the Jones ...
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46 views

What is the most rigorous book for knot theory?

I have read the knot book by Colin Adams and it has been very helpful with getting to grips with the basics. I struggle with showing things and proving things. I need a book with a lot of rigorous ...
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1answer
28 views

Knots winded around surfaces of a higher genus

There are many results for the knots winded around a torus, but what about knots winded around surfaces of a higher genus? Is there any classification of such knots? I would be glad to see any review ...
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0answers
16 views

unknotting number of surface-knot

Unknotting number of surface-knot is an invariant which is defined by the minimal number of 1-handles attached to obtain an unknotted over all possible surface diagrams representing it. A surface ...
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22 views

The HOMFLY polynomial existence proof

Theorem: There is a function \begin{align*} P:\{\text{Oriented links in $S^4$}\}\rightarrow\mathbb{Z}[l^{\pm{1}},m^{\pm{1}}], \end{align*} which satisfies the three axioms for the HOMFLY polynomial. ...
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1answer
54 views

Isotopic tori in $\mathbb{R}^4$

Intuitively it seems to me that two tori in $\mathbb{R}^4$ are isotopic to each other. By isotopic, I mean a smooth family of deformations beginning in one and ending in the other, and each member of ...
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0answers
30 views

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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2answers
61 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
45 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 ...
2
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1answer
34 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
105 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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0answers
39 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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1answer
39 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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0answers
24 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!
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40 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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1answer
29 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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0answers
39 views

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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0answers
54 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. ...
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0answers
31 views

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
44 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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37 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
44 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 ...
2
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0answers
57 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 ...
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2answers
34 views

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? ...
4
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0answers
56 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 ...
2
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1answer
83 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
39 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
31 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 ...
3
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3answers
42 views

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 ...
4
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2answers
113 views

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 $\mathbb{R}^3$: ...
2
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1answer
52 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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2answers
68 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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0answers
17 views

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 ...
5
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1answer
140 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
87 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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2answers
45 views

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.