For questions on knot theory, the study of mathematical knots

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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 ...
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59 views

More knots as crossing number increases?

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 ...
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Cohomology-Homology bilinear form of Seifert surfaces

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, ...
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1answer
56 views

Proof that knot genus is a knot invariant

I have a proof of the the following fact concerning knot genus, but I'm not sure that it's correct. If knot $J$ is isotopic to another knot $K$ then $J$ and $K$ have the same genus. Proof. Let ...
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Knot theory: Genus of a surface

Use Euler characteristic to determine the genus of the surface in Figure 4.24 in picture below. I am stuck with this question 4.10 from Colin Adams, the Knot Book.
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Crossing bound implies Reidemeister move bound?

In 1998 Galatalo established an upper bound on the number of Reidemeister moves needed to convert a diagram $D$ of the unknot into a trivial loop diagram. The upper bound is a function of $n$, the ...
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Seifert surfaces in Riemannian manifolds?

Does there exist an equivalent to Seifert surfaces for other Riemannian manifolds than $\mathbb{R}^3$? More precisely: Let $M$ be a simply-connected Riemannian manifolds and $K \subset M$ a (tame) ...
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122 views

Knot theory: Braids

Show by using a picture, that the two braids $\sigma_{i} \sigma_{i+1} \sigma_{i}$ and $\sigma_{i+1} \sigma_{i} \sigma_{i+1}$ are equivalent. This is 5.26 in knot book by Colin Adams. Need some ...
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the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of ...
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The math notation of this links? (connect sum of Hopf links)

We know the Hopf link owns the name of $2^2_1$ for Alexander–Briggs notations. (And there is another two component links is $4^2_1$.) I learned that "$4^3_1$ is not usually written as any three ...
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Alexander–Briggs notations for the links or knots of $N^3_m$

We can use Alexander–Briggs notations for the links or knots. For example, is three separate loops with no links. And there are many other examples of Alexander–Briggs notations for three ...
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Diagrams of links in public domain or licensed under Creative Commons

I'm writing a set of notes on topology that I'd like to share under the Creative Commons. Does anyone know where to find diagrams for links (not the Borromean link, I have that already) that are ...
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60 views

Is a knot $K$ and it's mirror image $^*K$ considered the same knot in terms of tabulating prime knots? If so, why?

I'm just wanting to confirm whether this is the case and why? Is it purely to do with the sheer number of knot projections that would have to be dealt with?
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Knot Theory: Mutations

Show that if we have three tangles as in Figure 2.33a, we can mutate several times in order to permute the tangles. Note that we can then permute n tangles in a row. This is from Colin Adams; The ...
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Hyperbolic distance

Find the hyperbolic distance between $(0; 0; 0)$ and $(0; 0; \frac12)$ in the Poincare model. Recall that the Poincare model deems $d(P_1; P_2)=\int\frac{2}{1-r^2}ds$. What about the distance between ...
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1answer
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Knot theory: pretzel knot

Prove that pretzel knot $K(p_1,p_2,p_3,\dots,p_n)$ with all $p_i >0$ is an alternating knot or link? I think since all $p_i$'s are positive, the sign has a lot to do with it but how to prove it is ...
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Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
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1answer
96 views

Knot theory question

Show that a (p,q) torus knot always has a projection with p(q-1) crossings. I can show an example using (2,3) has 4 crossings. I think there is something more to this. Help please
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1answer
168 views

How to draw stick trefoil knot

http://commons.wikimedia.org/wiki/File:Stick_number_trefoil.png I am interested in plotting the stick trefoil knot. I don't know where to start. I am looking for equations or co-ordinates of ...
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1answer
73 views

Why is there an ambiguity in Dowker's Notation for Composite Knots?

studying some knot theory and just had a question, wondering if anyone can clarify or shed some light: I'm reading The Knot Book by Colin C. Adams, and it says that Composite knots are not completely ...
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167 views

homeomorphism of cantor set extends to the plane?

Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ...
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1answer
60 views

Trivialization of the normal bundle of a knot

Let $ \phi $ be an embedding of $S^1$ in $ R^3$ or $S^3$. It is often mentionned (for instance when discussing framed knots) that one can choose a trivialization of the normal bundle $ \nu ...
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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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60 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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1answer
103 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
55 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
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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
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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
171 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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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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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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63 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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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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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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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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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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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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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
104 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
102 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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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
166 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
429 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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1answer
84 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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1answer
345 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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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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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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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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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 ...