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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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
359 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
556 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
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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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275 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 ...
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466 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
314 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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168 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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369 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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152 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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242 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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518 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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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 ...
2
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1answer
261 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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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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137 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
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2answers
249 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 ...
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5answers
826 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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482 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
523 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 ...
7
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584 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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186 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
270 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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211 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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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 ...
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5answers
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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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976 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
511 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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455 views

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 $0\leq k\leq n/2$. ...
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304 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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350 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
806 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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229 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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560 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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197 views

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 ...
2
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1answer
262 views

First cohomology of the space of long knots in R^4

Let's consider the space of long knots in $\mathbb R^n, n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I ...
10
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430 views

Traces on Hecke algebras and the Jones polynomial

In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr_z$ on the Iwahori-Hecke algebras $H(q,n)$ of type ...
12
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1answer
660 views

Fox differential calculus and the Alexander invariant of a link

I am teaching a course in knot theory, and I would like to describe the presentation of the Alexander module of a link obtained via Fox differential calculus. In doing this, I should prove the ...
17
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7answers
2k views

Is there a “knot theory” for graphs?

I think knot theory has been studied for quite a while (like a century or so), so I'm just wondering whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) ...
4
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234 views

Reshetikhin-Turaev and links with a distinguished component

Hi, This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question. Let $T$ be the category whose objects are ...
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193 views

A knot complexity measure

Construct a knot/link by fusing two n-tangles together. (A tangle matrix shows how this might look for tangles with 6 legs. But lets use 4 legs for a start as this is far simpler.) Now, any rational ...
5
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1answer
651 views

Kontsevich Integral without associators?

Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding ...
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321 views

A Category of Knot Diagrams

A brief explanation of my motivation before I ask my question. I am trying to understand Skein relations, the Jones polynomial, and their relations to Khovanov homology. To me, the natural setting to ...
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112 views

Spectral decomposition of R matrix -> Wenzl projectors?

Just curious: if you take a R matrix from knot theory and apply a spectral decomposition (see. e.g. my following post Matrix decomposition the other way) you'll get projectors: ...
6
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316 views

Number of the Reidemeister moves needed to transform one diagram into another one

A recent question Random Reidemeister moves to unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. ...
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Around the Montesino-Nakanishi 3-move

I have a few questions around the 3-move. I know it's NOT an unknotting move (but who needs knots with 20+ crossings anyway :-) by the recent proof of Przytycki. 1. In another paper about the third ...
6
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2answers
637 views

Random Reidemeister moves to unknot

Suppose one has a link diagram of the unknot, and applies random Reidemeister moves until the unknot is reached. Surely it requires an exponential number of moves, exponential in, say, the crossing ...
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1answer
491 views

Recovering the Alexander Polynomial from Ocneanu's HOMFLYPT

Let $H_q(n)$ denote the Hecke algebra associated to the symmetric group $S(n)$: this is the $\mathbb{Z}[q^{\pm 1/2}]$ algebra generated by $T_1, \ldots, T_{n-1}$ satisfying the braid relations along ...