**6**

votes

**1**answer

369 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

646 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

253 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

496 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

335 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

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**

votes

**2**answers

288 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

669 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

528 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

504 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**

votes

**1**answer

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**

votes

**5**answers

860 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

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

**0**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

376 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

591 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**

votes

**3**answers

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**

votes

**1**answer

280 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**

votes

**2**answers

478 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

327 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**

votes

**0**answers

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**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

567 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

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 ...

**2**

votes

**1**answer

267 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

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 ...

**1**

vote

**0**answers

139 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

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**

votes

**5**answers

848 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 ...

**8**

votes

**1**answer

490 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**

votes

**2**answers

526 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**

votes

**2**answers

593 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**

votes

**1**answer

188 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 $f(K)$ ...

**2**

votes

**1**answer

293 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**

votes

**1**answer

224 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). ...

**1**

vote

**0**answers

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**

votes

**1**answer

207 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**

votes

**5**answers

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**

votes

**1**answer

980 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 ...

**1**

vote

**1**answer

516 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 ...

**6**

votes

**0**answers

459 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 $0\leq k\leq n/2$.
...

**3**

votes

**0**answers

319 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**

votes

**2**answers

353 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**

votes

**1**answer

833 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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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**

votes

**1**answer

579 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 ...