**1**

vote

**1**answer

209 views

### Reference for a proof of the Dehn presentation

I would like a reference for a proof that the Dehn presentation is a presentation of the fundamental group of the knot complement in $\mathbb{S}^{3} $.

**3**

votes

**1**answer

317 views

### Can we get the HOMFLY polynomial for a torus knot from the Kauffman Polynomial?

This is essentially a yes/no/reference request question. I posted it on math.se and left it there for 5 days before posting here.
Let me first just ask my question: Is there a known relationship ...

**1**

vote

**2**answers

200 views

### A question on (1,1) bridge Knot

Hi, everyone. I am interested in the complement of (1,1) bridge knot in a lens space, $S^{3}$. Is there one (1,1) bridge knot in $S^{3}$ or lens space such that its complement is
hyperbolic?
...

**4**

votes

**1**answer

265 views

### Equivalent to Oriented knot complement conjecture

I would like to see why the following two statements in Kirby's list of problem are equivalent:
Statement 1:
If $K_1$ and $K_2$ are knots in a closed oriented 3-manifold $M$ whose complements are ...

**3**

votes

**1**answer

237 views

### A Degree of an Arbitrary Polynomial Knot

Here a degree of a polynomial knot is a minimal degree which can define a long knot. I would like to find out how this degree can be bounded below, according to the number of crossing points, for ...

**5**

votes

**3**answers

585 views

### Links with same Jones polynomial

Is there anything known about when two links have the same Jones polynomial (beyond a calculated list of small actual examples)? The first thing I would try is to compute the (formal - you would have ...

**0**

votes

**1**answer

386 views

### How many 3-flip Mobius strip knots are there?

Take a clock-wise 3-flip mobius strip,
Cut it down the middle and then let the ribbon cross itself 6 times.
This forms a framed knot of which there are many.
Get the knot diagram.
I've found ...

**1**

vote

**0**answers

165 views

### Things you can do with the self-writhe

I hope "self-writhe" is the established word. (0 for link-crossing, otherwise identical to writhe +1 or -1) I bet the following is known: Take some crossing of a link with self-writhe $w_a$. Flip it ...

**3**

votes

**1**answer

331 views

### Unknotting number and crossing number

It is well known that if $c(K)=2n+1$, then $u(K)$ is less than $n+1$. It can not be sharper because of the trefoil knot. On the other hand, if $c(K)=2n$, then similarly we have $u(K)$ is less than ...

**6**

votes

**1**answer

859 views

### What is knot contact homology?

Recently, it was conjectured by the paper of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed ...

**0**

votes

**1**answer

270 views

### Thurston-Bennequin number vs. checkerboard coloring difference

For an alternating knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1
kinks exist (K has ...

**4**

votes

**1**answer

366 views

### Problems about the Estimate the Unknotting Number

For the definition of unknotting Number, you can assess http://www.popmath.org.uk/exhib/pagesexhib/unknum.html
My question is:
For given a knot K, let n be the crossing number of K, is their any ...

**7**

votes

**0**answers

242 views

### What is the historical connection between Zeeman's twist spinning and Fox's Examples?

Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through ...

**1**

vote

**1**answer

168 views

### Generalizing the Reshitikhin-Turaev construction possible?

OK, I have to ask a dumb question again: Where do Lie groups enter in the
construction of the Reshitikhin-Turaev invariant? The parts of the proof I
understand are that 6j symbols take care of ...

**34**

votes

**2**answers

2k views

### Knot security (When to trust your life with a knot)

This question is related to a a question about self-tightening knots.
I am supervising a senior thesis and my student is interested in knots. My student is also a rock climber and has an ...

**1**

vote

**1**answer

339 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

594 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

395 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

190 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

363 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

507 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

263 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

663 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

887 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

305 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

594 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

598 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

330 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

273 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

342 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

860 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

385 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

412 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

112 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

406 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

727 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

274 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

520 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

373 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

178 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

411 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

803 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

731 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

546 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

268 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

235 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

249 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 ...

**27**

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

413 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 ...