For questions on knot theory, the study of mathematical knots

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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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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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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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Subsets of immersions that are embeddings

Let $X$ be a manifold and let $Y$ be a submanifold, possibly with boundary. I am dealing with a situation where $f:X\to \mathbb{R^3}$ is an immersion, but $f\vert_Y:Y\to \mathbb{R}^3$ is an embedding. ...
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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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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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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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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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branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
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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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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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2answers
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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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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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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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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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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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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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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
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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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1answer
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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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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
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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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108 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
55 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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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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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
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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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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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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
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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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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
215 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
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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 ...
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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
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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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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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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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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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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 ...