**2**

votes

**1**answer

192 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

827 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

224 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

365 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

575 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

214 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

277 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

470 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

316 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

170 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

372 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

154 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

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**

votes

**1**answer

535 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

311 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

263 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

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**

votes

**2**answers

256 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

836 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

486 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

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 ...

**7**

votes

**2**answers

586 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

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 $f(K)$ ...

**2**

votes

**1**answer

276 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

218 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

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**

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

977 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

513 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

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

**3**

votes

**0**answers

310 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

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**

votes

**1**answer

816 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

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**

votes

**1**answer

566 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**

votes

**0**answers

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**

votes

**1**answer

264 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**

votes

**1**answer

438 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**

votes

**1**answer

673 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**

votes

**7**answers

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**

votes

**0**answers

235 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 ...

**1**

vote

**0**answers

194 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**

votes

**1**answer

653 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 ...

**1**

vote

**0**answers

327 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 ...

**1**

vote

**0**answers

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: ...