For questions on knot theory, the study of mathematical knots

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1answer
63 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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3answers
378 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
129 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
69 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
112 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
111 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
40 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
174 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
435 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
86 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
361 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
77 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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30 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
69 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
45 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
60 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
49 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
51 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
291 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
91 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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0answers
34 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
72 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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39 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
16 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 $p(L)$ ...
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1answer
58 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
53 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
115 views

Knot complement conjecture in solid tori

Has the knot complement conjecture been proven for knots in solid tori?
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0answers
80 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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1answer
144 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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56 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
42 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 ...
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1answer
82 views

Genus of a link

In knot theory, we know that same linking number cannot distinguish two different knots/links. For example, whitehead link(linking number$=0$) and unlink of 2 components (linking number$=0$) but ...
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42 views

Relationship between Kauffman and HOMFLY polynomials

If we let $F_{L}(t)$ denote the Kauffman polynomial and $P_{L}(x,y)$ denote the HOMFLY polynomial, then we can obtain the Kauffman polynomial from the HOMFLY polynomial using the following ...
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0answers
63 views

Mirror images of knots and Kauffman and HOMFLY polynomial

Let $K$ is a knot and let $\bar{K}$ be the mirror image of $K$. I want to confirm this relationships. Let $f_K(t)$ be the Kauffman polynomial of $K$. To get the mirror image we swap every right ...
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0answers
46 views

Name of a link invariant?

Below I will describe a link invariant, denoted by me as $inv(L)$. Has anyone encountered this invariant in the literature? If so, what is its name? Also, any references to papers or books that ...
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0answers
64 views

Reidemeister Moves

In knot theory, two links are equivalent if and only if they can be deformed from one to another by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know ...
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0answers
50 views

Conway polynomial of the unknot

I was trying to follow along with Wikipedia's basic computation of the Conway polynomial of the trefoil knot (http://en.wikipedia.org/wiki/Knot_theory#Knot_polynomials), but I got sidetracked by ...
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1answer
232 views

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ...
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0answers
108 views

Legendrian Isotopy of Knots can be extended to an ambient Contact Isotopy

I am attempting to understand a proof that an isotopy of two Legendrian knots $L_0$ and $L_1$ in a closed contact manifold (M,$\xi$) can be extended to an contact isotopy $\phi$ of M such that $\phi_0 ...
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0answers
84 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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0answers
59 views

What is a 2-surgery on a disk?

I am confused by a certain point in Scharlemann's paper "Sutured Manifolds and Generalized Thurston Norms", which seems important enough to not just skip it. I mean the "2-surgery on disks" in the ...
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20 views

Determine the multiplicity of knots for a graph

Here are my two questions: Given a finite connected non-oriented planar graph, is there a way to determine whether or not it is possible to derive a single non-trivial knot diagram from this graph, ...
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2answers
98 views

Recommended books on knot invariants

I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd ...
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1answer
110 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
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1answer
50 views

Req. for Definition:Twisting Number of Curve in Contact Structure

All: I'm reading a paper that makes mention of the twisting $tw (\gamma,S) $ , where $\gamma$ is a simple, closed Legendrian curve in a surface $S$ , and $S$ is embedded in a contact 3-manifold ...
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1answer
177 views

Why is the Whitehead double of a knot always prime?

I was looking for a proof that there are infinitely many prime knots and one said "take your favorite (prime) knot and consider all its Whitehead double", implying that all Whitehead doubles of a ...
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0answers
91 views

On the definition of Fox derivative

I am reading An Introduction to Knot Theory by W.B. Raymond Lickorish. In Chapter 11 the motivation for the Fox derivative is mentioned. I understand why the contribution of the occurrence of $x_j$ in ...
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0answers
348 views

Wirtinger Presentation for the Figure eight knot, Rolfsen exercise

I have been working through Rolfsen's "Knots and Links" and have found myself frustrated by exercise 4 on page 58. It concerns the Wirtinger Presentation of the figure eight knot, where the ...
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2answers
302 views

Total mean curvature of an immersed torus.

How to prove that the total mean curvature of an immersed torus of $R^3$ such that has nontrivial self-intersection must $> 8 \pi$? The definition of total mean curvature is the integral of $H^2$ ...
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1answer
487 views

Complement of figure-8 knot

I am reading W. Thurston's famous "3-dimensional Geometry and Topology", but I am stuck at the point where it is said that gluing two tetrahedra in an appropriate way give you the complement of the ...