For questions on knot theory, the study of mathematical knots

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119 views

Wedding Vows puzzle

My father came up with a puzzle and dared me to solve it. I could solve it by trial and error, but I rather want to solve it mathematically. It is the so called "Wedding Vows puzzle" where you have to ...
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2answers
51 views

Does Reidemeister's Theorem apply to Links?

Reidemeister's Theorem states: Two knot projections $K_{1}$ and $K_{2}$ are equivalent if and only if $K_{2}$ can be obtained from $K_{1}$ by a sequence of Reidemeister moves. Does this ...
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70 views

Do the composition of two Knots always yield a distinct knot (ignoring orientation)?

I would greatly appreciate if I could get some help in clarifying my understanding. (This is a special topic I am studying as a 2nd year University student - I haven't taken topology yet - so please ...
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1answer
46 views

Bridge Number , Knot Theory

I had been reading some knot theory lately and got to know about a whole classification of 2-bridge knots , does their exist any such extensive study over 3-bridge knots?
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1answer
32 views

Example of using torus knots in experimental science

Can anyone give an example of using the theory of torus knot in experimental science? Thanks in advance!
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1answer
42 views

Applying Khovanov homology to two different non-trivial diagrams of the unknot

I'm attempting the calculate the Khovanov homology of the unknot using the figure eight diagram of the unknot with exactly one crossing going from top left to bottom right as shown below. I also ...
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1answer
154 views

How to ensure Topological Correctness

Question: I read through an enormous amount of material on topology and knot-theory in wikipedia, but I still am stuck at the following fundamental problem: Given two representations of closed ...
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0answers
27 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 ...
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1answer
50 views

Knot theory: showing that the ambient isotopy relation is symmetric

My apologies for this rather elementary question, but here goes: I didn't have any trouble figuring out how to prove that the 'standard' homotopy/isotopy definitions give rise to an equivalence ...
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1answer
51 views

Infiniteness of a knot energy

I am going through the paper, Physical Knots, by Jonathan Simon. Initially, on page 10, he describes the electrostatic energy of two charged stick, X and Y in space as follows. $$ \int_{x \in X} ...
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120 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 ...
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55 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): ...
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1answer
46 views

changing one crossing in (2,n) torus knot

I want to check whether my guess is true or not: changing one crossing in torus knot (2,n) gives a torus knot (2,n-2)? Is that true By changing crossing I mean exchanging over and under arcs. Thank ...
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24 views

coloring torus knot (2,n) diagram

Is it true that a torus knot (2,n) diagram is always colorable by a dihedral quandle of order n for any positive integer n >2? I know the above statement is true if n is prime what about the other ...
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3answers
130 views

Framing Integer associated with a Framed Knot/Link

I have been looking for a clear definition of the n-framing of a knot unsuccessfully. A framing here refers to a choice of homeomorphism between a solid torus neighborhood (a.k.a, tubular ...
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0answers
102 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 ...
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2answers
54 views

Prove that an infinitely long rope can only form slipknots

I've heard that an infinitely long rope can only form slipknots, is that true, and is there a simple proof/obvious counterexample? Answers requiring no preliminary knowledge about topology would be ...
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1answer
86 views

Skein Tree: Conway Polynomial

I am trying to learn about the skein relation, but I don't understand what is being done here. Can anyone help me with this? And how is $1+z^2$ as the final result obtained? Additional: This is the ...
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1answer
67 views

Knot Theory: Smooth vs Polygonal

So I will be giving a talk about knot theory and was wondering why would one study knots from a graph theoretical perspective, i.e a collection of edges and vertices? Is this just a preference? Is ...
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1answer
29 views

Minimizer and invariance of normal projection energy of a knot

While reading the paper, A simple energy function for knots, I understand that the authors have proved the two conditions of the first page for the normal projection energy of a knot. But I failed to ...
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1answer
110 views

Will a knot tied in a hanging, frictionless rope slip out under the force of gravity?

I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless ...
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1answer
386 views

Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which ...
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0answers
47 views

Knot Theory: Conway Skein Relation

I am trying to figure out what exactly is this proposition proving. Is it saying take the trivial link and attach u trivial links to it ( like the borromean rings)? Or are we just taking u copies of ...
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1answer
63 views

Knot Theory: Calculating the Alexander Polynomial

I am going to be giving a talk about knot thoery in a few weeks and I will be discussing different knot invariants-one of which being the alexander polynomial I am having a problem understanding how ...
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26 views

The number of colorings of given knot diagram

A colorings of knot diagram is a function from the set of arcs in the diagram to a given quandle such that a*b=c holds at each crossing, where a (resp. c) is the under arc that is behind (resp. in ...
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1answer
194 views

How to reason about disentanglement “tavern” puzzles?

It took me an embarrassingly long time to remove the ring from this rigid structure: What math could I use to solve similar puzzles? Topology and knot theory seem helpful, but I don't think they ...
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2answers
59 views

example of knot diagram colored by dihedral quandle of non-orime order, if any

Is there a known example of knot colored by a dihedral quandle of non-prime order, for example the diherdral qunadle of order 4, 6 or 12.
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28 views

Coloring knot diagram obtained from colored one by applying crossing change to one crossing

Suppose $K_1$ is a knot diagram colored by a dihedral quandle $R_n$ of order $n$, By applying crossing change (exchanging over and under arcs) to one crossing in $K_1$, we obtain a new diagram let us ...
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1answer
58 views

Link diagrams and Reidemeister moves

I am studying Knots on "Algebraic Graph Theory" written by Godsil & Royle. They state the following theorem: $\underline{Theorem}$ Two link diagrams determine the same link if and only if one can ...
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0answers
35 views

double decker set in a surface-knot

Surface-knot is an embedded surface in $\Bbb{R}^4$. Project the surface in $\Bbb{R}^3$ gives the surface diagram with set of singularity points consists of double points, triple points and branch ...
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2answers
48 views

Uniqueness of rotational symmetry of a link diagram

How can I prove that a connected link diagram can only admit up to one axis perpendicular to the plane through which rotational symmetry lies, i.e there aren't rotational symmetries through different ...
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1answer
41 views

Link complements in $\mathbb{R}^{3} $ and $S^{3} $

What's the difference between a link complement in $S^{3} $ and a link complement in $R^{3} $? Are they homeomorphic?
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0answers
12 views

Regular projection of link and a disk

Let $K$ be a link in $S^{3}$ and consider an associated link projection $p: S^{3} \rightarrow S^{2}$. Let $D$ be a solid disk in the regular projection. Is the preimage $p^{-1} (D) $ a ball? Or a bit ...
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31 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. ...
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1answer
163 views

Relation between the braid group and the mapping class group of the plane

According to the following link, page 248, the braid group modulo its center is isomorphic to the mapping class group of the $N$-times punctured plane, i.e. $B_N/Z(B_N)\cong M_N(\mathcal(R)^2)$. Could ...
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1answer
43 views

A generalization of the connected sum of links

A connected sum of two links $K$ and $L$ involves cutting a segment in each link and joining them up as illustrated in the top diagram, the connected sum of two trefoil knots. Is there any ...
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21 views

Relationship between the set of all canonical knots and the set of all knot genuses

What is the relation between the set of all canonical knots and the set of all knot genuses? I understand that there is at least not bijection between them.
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0answers
26 views

Fundamental group of the complement of $\operatorname{Wh}(\operatorname{Bor})$?

It is well known that $\operatorname{Wh}(\operatorname{Bor})$ link (That is, untwisted Whitehead double of Borromean rings with positive clasps, say) is very interesting. Are there any easy way to ...
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1answer
57 views

Moves for regular homotopies of immersions of $S^1$ in the plane

What is a set of moves to combinatorially describe regular homotopies of (smooth) immersions $S^1\to \mathbb R^2$?
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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 ...
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0answers
14 views

Induced cyclic ordering in link diagrams

Let $L$ be a link in $\mathbb{R}^{3}$, and $p : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ a regular projection (i.e. injective everywhere, except at a finite number of crossing points) and so $p(L)$ ...
3
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1answer
41 views

Unknotting number formally?

I am reading Colin C. Adams's very nice but not always rigorous "The Knot Book" right now. How does one formalize the unknotting number? (For example, is some restriction on embeddings ...
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1answer
43 views

What does the tensor product in the definition of Combinatorial Floer knot homology look like?

I am working on a project that involves summarizing the article A combinatorial description of knot Floer homology (http://arxiv.org/abs/math/0607691) and doing some example computations with the ...
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1answer
93 views

Knot complement conjecture in solid tori

Has the knot complement conjecture been proven for knots in solid tori?
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0answers
71 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 ...
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0answers
18 views

Torsion energy as a component of physical energy of a knot

I am going through the paper, Recognizing knots Using Simulated Annealing by Ligocki and Sethian. This paper uses simulated annealing to solve the KNOT GENUS problem. It has used the concept of ...
2
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1answer
91 views

Dehn and Wirtinger Presentations of Knot Groups and their connection

I'm currently working through N.D. Gilbert and T. Porter's Knots and Surfaces. In it the idea of a Wirtinger presentation and a Dehn presentation for a group associated with a given knot is ...
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40 views

Invariance of the physical energy of a knot over Möbius transformations

I am going through the paper, Recognizing knots Using Simulated Annealing by Ligocki and Sethian. This paper uses simulated annealing to solve the KNOT GENUS problem. It has used two different ...
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0answers
55 views

What is the type of $u$ in this definition of knot?

I am going to quote from the second paragraph of the Introduction of Möbius Energy of Knots and Unknots by Michael H. Freedman, Zheng-Xu He and Zhenghan Wang. Let $\gamma = \gamma (u)$ be a ...
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1answer
36 views

Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...