Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.

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AJ conjecture for links

Garoufalidis proposed a conjecture on $q$-difference equations for the colored Jones polynomials of knots. \begin{equation} \hat{A}_K(\hat{l},\hat{m};q)J_n(K;q)=0 \end{equation} where the actions of ...
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2answers
252 views

Is there a notion of “ribbon 2-category”?

It there some notion of ribbon 2-category, which would allow for, say, talking about the Seifert surface of links (which is a 1-morphism in some ribbon category) as a 2-morphism in the category? ...
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4answers
495 views

Stick knot questions: simple but may not be easy

I have a few questions about nonplanar "stick circuits" (or hexagons and higher $n$-gons) that you might be able to help with: (I know that $n=6$ is the minimum number of points to form a stick ...
3
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1answer
331 views

Framings in the definition of Reshetikhin-Turaev TQFT

I posted the following question at Mathe Stack Exchange.link text But it has not yet answered. I am sorry if you check both sites but I also want people here to look at this problem. I am studying ...
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0answers
152 views

link group of the trivial $n$ component link

Currently I am reading Milnor's paper "Link Groups". In the paper he defines the link group, for every $n$ component link $L$, as a certain quotient of the fundamental group $\pi_1 (S^3 \setminus L)$. ...
4
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2answers
275 views

Computational complexity of Knot polynomials

What's known about computational complexity of different types of knot invariant polynomials? For example, Evaluating Jones Polynomial is known to be #P hard. Is there any reference that surveys such ...
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3answers
665 views

On connection between Knot theory and Operator algebra

What exactly is the connection between knots and operator algebra? I heard that Jones established such a connection while discovering the celebrated Jones Polynomial. Now Jones Polynomial is ...
7
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1answer
520 views

Rational homology spheres and knots

It is known that the boundary of a branched double cover of the four ball branched over a surface bounded by a knot in is a rational homology $3$-sphere. Two questions come to my mind simply out of ...
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1answer
498 views

Integer matrices with a strange divisibility property

Fix an integer $n$. What can you say about a (not necessarily square) matrix $A$ with integer entries that has the property that for any $k$, every $k\times k$ minor of $A$ is divisible by $n^{k-1}$? ...
2
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1answer
193 views

Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots, especially minimum bend drawings. An orthogonal drawing employs segments parallel to the axes of a Cartesian coordinate system. A bend is a ...
10
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5answers
850 views

Best Computational Knot Invariants

My apologies if this is too closely related to this closed post. I have been collaborating with a physicist looking at long polymer chains. These chains form knots with 2D projections having on the ...
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1answer
226 views

Source on the proof that codimension 2 is sufficient for knottings?

Hi all. I'm not even sure that this is a theorem, but a while ago I heard a topologist friend of mine (whom I haven't been able to reach) saying that given a continuous embedding $f:M^{n}\to N^{r}$ ...
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1answer
246 views

Embedding of $T^{2}$ on $S^{1}\times S^{2}$.

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be a embedding map and $i_{*}:\pi(T^{2})\rightarrow \pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times ...
26
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1answer
1k views

Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$. As a corollary of something else I was playing around with, I recently ...
5
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2answers
373 views

Tangled Knot Function

I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$ that has these properties: (1) When iterated $n$ times starting from some $p$, connecting the points in order with segments and closing ...
5
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2answers
587 views

Applications of the knot theory to biology/pharmacology ?

What are the applications of the knot theory to biology/pharmacology ? I guess there should be some, since proteins are quite long and probably some of their properties are related whether they are ...
3
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3answers
219 views

slice knots, what does the locally flat condition say?

I am studying slice knots, so for example they say the cone on a trefoil knot can be embedded in D^4 but it is not locally flat at the vertex of the cone. What I do not understand that I think every ...
5
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1answer
278 views

Jones(unlink)=phi

Somewhat nebulous question: there are many well known "special" values of the Jones polynomial, especially those at roots of unity. I always run into one that has unlink value $\phi$ (golden mean) and ...
7
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2answers
472 views

Maslov index and heegard floer homology

I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this expository paper which was working quite well until I reached the actual definition of the differential. ...
4
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1answer
321 views

Khovanov-Rozansky $sl_2$ homology and the “original” Khovanov homology.

I'm trying to understand the connection between Khovanov's original link homology and the $sl_2$ version of Khovanov-Rozansky homology. They both categorify the same link polynomial, but is there a ...
3
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0answers
172 views

Constructing Markov traces simply

Short version: I wondering how to simply check if a proposed Markov trace, $\phi$ had the correct property using techniques similar to those from the Akutsu-Wadati 1987 paper `Exactly solvable models ...
9
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1answer
379 views

Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively

I'm interested in a complexity question related to problems like the slice-ribbon problem. To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...
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155 views

Finding a ribbon graph for a mapping class group action

Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$. This action $\epsilon$ is ...
2
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1answer
247 views

Computing an Invariant for Knots via Braid Words?

I've been reading up on Knot Theory (which is not my area of expertise) and am stuck in the following bit of logic: Statement 1: Every knot can be represented as a braid. Statement 2: There's a ...
4
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1answer
555 views

Closed formula for colored Jones polynomial of the trefoil? (reference request)

(EDIT: Powers of $q$ in the formula corrected.) I've been doing some computations with skein modules, and I found the following formula for the N-th colored Jones polynomial of the trefoil: ...
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316 views

What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds ?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
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1answer
266 views

A kind of foliantion on figure eight knot complement

Let $N$ be the figure 8 knot complement, What we can say about such kind of dim 2 foliation $F$ on $N$: (1) no Reeb (2 dim); (2) $F$ intersect transversly with $\partial N$ is $n$ pareller Reeb (1 ...
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1answer
270 views

A special ribbon graph presents a cylinder.

I am reading "Quantum Invariants of Knots and 3-Manifolds" by Turaev. I have a dificulty to understand the proof of Lemma 2.6 on page 172. The lemma says that a special ribbon graph drawn on page 167 ...
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0answers
138 views

Is there a two-variable E8 polynomial? (Conjectural or proven)

On MO I learnt about the two-variable E7 polynomial (status: conjectural). What about a two-variable E8 polynomial? I have reasons to believe such a thing exists too, but I do magic, not math, so my ...
9
votes
2answers
257 views

Higher homotopy groups of slice disk complement

Let $K \subseteq{\mathbb{S}}^3=∂\mathbb{D}^4$ be a non-trivial slice knot, i.e. $K$ bounds a slice disk $\Delta$ in $\mathbb{D}^4$. Let $N(\Delta)$ be a regular neighborhood of $\Delta$ in ...
10
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5answers
842 views

What is the metamathematical interpretation of knot diagrams?

I am not a geometric topologist, but from looking over papers in the field, it's clear that knot diagrams are a major tool and we know how to use them in a way that is rigorous and trustworthy. My ...
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1answer
488 views

Knot Invariants from Twisted Quantum Doubles

In "Topological Gauge Theories and Group Cohomology", Dijkgraaf and Witten construct a 3-manifold invariant from a finite group $G$ and 3-cocycle $\omega$. I would think there is also an associated ...
4
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2answers
525 views

Unknotting knots in 4D

Suppose one has a knot $K$ embedded in $\mathbb{R}^3$; but view $\mathbb{R}^3$ as a 3-flat in $\mathbb{R}^4$. Of course $K$ is not a knot in $\mathbb{R}^4$. I am wondering if there has been any study ...
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2answers
590 views

Quantum E6/E7 knot polynomials

Has anybody seen seen quantum knot invariants associated to (E6, 27) or (E7, 56) worked out in the literature? Even for just simple knots like the trefoil or figure-8? I suspect these haven't been ...
3
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1answer
187 views

Knot symmetries and the Alexander polynomial

Let $K\subset S^3$ be a knot. Suppose there is an involution, $f$, of $S^3$ such that $f(K)=K$, and the fixed points of $f$ do not lie on $K$ itself. Furthermore assume that the orientations of $f(K)$ ...
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1answer
289 views

Why Tristram-Levine signature jumps at the zeros of alexander polynomial?

It seems easy but I can't prove it. Can anyone give proof or reference?
4
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1answer
222 views

What vector space does the Kauffman bracket skein algebra of FxI act on?

The Kauffman bracket skein module $K_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). ...
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0answers
154 views

Measuring the complexity of a knot by minimum number of simplices to tile the complement

This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb ...
4
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1answer
206 views

Growth of knots possible with rope of length L

What is the asymptotics (in L) for the number of topologically different knots possible using a perfectly flexible, non-selfintersecting rope of length L and radius 1? (With ends glued together after ...
11
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5answers
1k views

Why “Categorify”? Relating to link/knot homologies…

Hey Everyone! So I am new blood in the topic of Khovanov Homology and related topics. According to my basic reading the idea is to get the Jones polynomial as the Euler Characteristic of a certain ...
9
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1answer
979 views

Self-tightening knot

Is there a way, for some finite L>1, to tie two pieces of rope together, such that any finite force is not enough to pull them apart? The type of rope I have in mind is something like cylindrical ...
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1answer
515 views

Knots that are not knots [closed]

1) Classic Knotting problem: Classify embeddings of circle into 3D Euclidean space up to isotopy. http://en.wikipedia.org/wiki/Knot_theory 2) General topological knotting problem: Classify embeddings ...
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Does anyone know this sequence of polynomials?

A referee on a paper of mine showed me the following recurrence for polynomials $P_{n,k}\in\mathbb Q[q,q^{-1}]$ for $n\geq 0$ and $0\leq k\leq n/2$. ...
3
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0answers
316 views

Connected Sum Decomposition of a Knot

Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?
3
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2answers
351 views

Is there a known method for finding the minimum bridge index of a knot?

It is easy to establish an upper bound $n$ for the bridge index of a knot by producing a diagram with the knot in $n$-bridge position. Is there a known method to produce a reasonable lower bound ...
10
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1answer
826 views

Is there a periodic table for knots?

When I see knot tables, I have two feeling: ah, it's beautiful, and... painful. I don't see how knots are ordered in the knot table, the way to go from one knot of a certain crossing number to ...
3
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1answer
214 views

a special type of 2 component link complement

It is well know that a 2 component link complement may doesn't detect the link type. My question is whether the following type of 2 component links detect their link types? Such a link is composed ...
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230 views

Knots that turn around an axis [closed]

Take a thick cord (the alim cord of your laptop or your mouse cord for example) and wrap it around your hand (or finger) turning always in the same direction but possibly knotting it. Then try to ...
4
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1answer
574 views

Does there exist infinitely many prime knots?

I'm not a topologist and I just saw the definition of prime knot a while ago. Today I'm somewhat supprised to realize that I don't even know if there are infinitely many prime knots? If this ...
3
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0answers
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Knot polynomials of non-crystallographic Coxeter groups?

I learnt that the Coxeter groups have a few members more than the classic simple Lie groups: $H_3, H_4$ and $I_2(p)$. Is there a Reshetikhin-Turaev invariant for those, too? If not, where does the ...