**1**

vote

**1**answer

337 views

### Can Reidemeister 3 be weakened?

If you take the diagram of the Reidemeister 3 move and "shortcircuit" two ends, you get (click http://imgur.com/kRvZa if Imgur hotlink doesn't work):
...

**11**

votes

**1**answer

580 views

### How much do homological knot invariants improve the classification problem of knots?

The mutation operation in knots appears to be detected by the Floer homological invariants. See the papers by Ozsvath, Szabo and by Baldwin, Gillam. In addition, the Khovanov homology turns out to be ...

**9**

votes

**2**answers

385 views

### $6j$-symbols for $U_q({\mathfrak{sl}}_n)$ and colored HOMFLY polynomials

Explicit expression of quantum $6j$-symbolos for $U_q({\mathfrak{sl}_2})$ have been known due to the work of Kirillov and Reshitikhin.
My Question:
How much are known about quantum $6j$-symbolos ...

**2**

votes

**1**answer

187 views

### Application of a quandle cocycle invariant for virtual knots

In knot theory,
a quandle cocycle invariant was defined.
Moreover, to virtual knot theory it was generalized by avoiding for virtual crossings.
Question
Are there many application of a quandle ...

**2**

votes

**2**answers

359 views

### Infinite knot composed of parallel helices

I am wondering if there is a developed theory of "infinite knots" that could capture
this object, and tell me something of its knot properties.
Imagine vertical helices in $\mathbb{R}^3$, each ...

**9**

votes

**3**answers

501 views

### Circle-arc number of a knot

I would like to build knots in $\mathbb{R}^3$ from arcs
of unit-radius (planar) circles, joined together at points where
the tangents match. Thus the knot will have
curvature $1$ at all but the ...

**0**

votes

**1**answer

252 views

### Validity of generalized Reidemeister moves for a virtual knot

I am studying virtual knot theory.
A virtual knot is a knot diagram with real or virtual crossing information.
The equivalence relation includes generalized Reidemeister moves.
There are premitted ...

**8**

votes

**4**answers

643 views

### Knot diagrams, sets of moves and equivalence relations

Short version: Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams?
Yes, I understand that ...

**5**

votes

**6**answers

868 views

### Knot theory without planar diagrams?

I remember reading somewhere that there are just a few contributions to knot theory which do not involve knot diagrams. Hence my question:
Does anybody know about papers concerning knot theory which ...

**4**

votes

**0**answers

302 views

### On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials.
Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...

**9**

votes

**0**answers

570 views

### Categorification of finite type invariants

Recall that finite type invariants are those numerical knots invariants vanishing on knots with enough singularities. The value of an invariant on a singular knots is computed using the Vassiliev ...

**10**

votes

**3**answers

572 views

### Knot theory question: bridge number vs. min generators of fundamental group of complement

Given a knot in the 3-sphere in Bridge Position you can find a presentation for the fundamental group of the complement (a Wirthinger presentation) containing $n$ generators and $n-1$ relators, where ...

**6**

votes

**1**answer

325 views

### Computations and applications of Khovanov's functor valued invariant of tangles

Soon after his famous paper "A categorification of the Jones polynomial", Khovanov
introduced a "bordered" version. His theory assingns to every oriented even tangle
a complex of (H_n,H_m)-bimodules, ...

**3**

votes

**1**answer

268 views

### On expressions of colored Jones polynomials

In the paper by Masbaum, it was shown that the colored Jones polynomials for a twist knot $K_p$ can be written as
\begin{eqnarray}
J_{n}(K_p;q)&=&\sum_{k=0}^{\infty} {\cal C}_{K_p}(k)
...

**5**

votes

**1**answer

337 views

### Euclidean symmetries of torus links in R^3

I have a question about whether Ryan Budney's question:
Torus knots in Euclidean space -- a symmetry argument
can be extended to links. He asks:
Suppose you have a $(p,q)$ torus knot $K$ in ...

**1**

vote

**1**answer

824 views

### Homology and homotopy type for knot complements

I'm trying to understand a paper by Kawauchi and Matumoto, 'An estimate of infinite cyclic coverings and knot theory.' In one of their proofs they have a manifold $E$ which is the complement of a ...

**7**

votes

**2**answers

375 views

### High-dimensional ribbon knots

Let us suppose that we have a ribbon embedding $S^n \rightarrow S^{n+2}$ for $n\geq 3$. Call this knot $K$. By a theorem of Levine (and Trotter for $n=3$ I believe) we know $K$ is unknotted if the ...

**7**

votes

**1**answer

410 views

### Resemblance between Birman-Murakami-Wenzl algebra representations and the Lawrence-Krammer representations

At the end of Stephen Bigelow's paper "Braid Groups are Linear", he mentioned that there is a striking resemblance between the matrices of the Lawrence-Krammer (LK) representations and "those of a ...

**1**

vote

**0**answers

110 views

### Minimal piecewise-linear knot diagram [closed]

I'm looking for an answer to the following question:
Given a knot in $\mathbb{R^{3}}$ can we find a piecewise-linear diagram of it wich is minimal (has a minimal number of verticies)?

**6**

votes

**1**answer

399 views

### Examples of calculations of Turaev-Reshetikhin TQFT of cobordisms with boundaries have genera greater than 1

I am studying Turaev-Reshetikhin TQFT. I describe the definition of the invariant $\tau(M)$ of a cobordism $(M, \partial_{-}M, \partial_{+}M)$ in the previous question breifly. Framings in the ...

**7**

votes

**2**answers

711 views

### 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 ...

**4**

votes

**2**answers

270 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?
...

**4**

votes

**4**answers

518 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**

votes

**1**answer

369 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 ...

**0**

votes

**0**answers

172 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**

votes

**2**answers

374 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 ...

**11**

votes

**3**answers

768 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**

votes

**1**answer

679 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 ...

**12**

votes

**1**answer

541 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 $n^{k-1}$? ...

**3**

votes

**1**answer

264 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**

votes

**5**answers

1k 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 ...

**1**

vote

**1**answer

233 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 $f:M^{n}\to N^{r}$ ...

**0**

votes

**1**answer

247 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**

votes

**1**answer

2k 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**

votes

**2**answers

407 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**

votes

**2**answers

790 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 ...

**4**

votes

**3**answers

280 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**

votes

**1**answer

288 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 ...

**9**

votes

**2**answers

548 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**

votes

**1**answer

396 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 ...

**4**

votes

**0**answers

191 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**

votes

**1**answer

411 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 ...

**0**

votes

**0**answers

161 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**

votes

**1**answer

261 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**

votes

**1**answer

678 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:
...

**8**

votes

**0**answers

355 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 ...

**2**

votes

**1**answer

286 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 ...

**2**

votes

**1**answer

290 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 ...

**1**

vote

**0**answers

147 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

**2**answers

299 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 ...