**9**

votes

**0**answers

290 views

### Is the connected sum of knots an isometry?

Take $X$ as the set of knots in the 3-sphere (i.e. smooth embeddings of $S^1$ in $S^3$ up to smooth isotopy), endowed with the Gordian distance $d$.
For a fixed knot $K$ we can define the map ...

**1**

vote

**1**answer

454 views

### Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...

**7**

votes

**1**answer

443 views

### Diagrammatic proof of unique prime decomposition of knots

Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident ...

**5**

votes

**1**answer

192 views

### Locally flat non-smooth discs

There are many knots (e.g. the $P(-3,5,7)$-pretzel knot) that are topologically, but not smoothly slice; "topologically" slice means that there is a locally flat embedding of a disc into the 4-space, ...

**10**

votes

**2**answers

338 views

### Genus one fibered links

It is well-known that the only genus one fibered knots are the trefoil and the figure-eight. On the other hand, there exist infinitely many fibered links for any fixed higher genus.
My question is ...

**1**

vote

**0**answers

97 views

### Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...

**1**

vote

**0**answers

92 views

### Can all knot polynomials derived from Skein relations be categorified?

Can all knot polynomials derived from Skein relations be categorified?

**5**

votes

**1**answer

177 views

### Seifert genus of the lift of a knot in its cyclic branched covers.

I was wondering if there are any known examples of knots $K$ in $S^3$ with Seifert genus $g$ so that the lift of $K$ sitting inside its $n$-fold cyclic branched cover bounds an embedded surface of ...

**2**

votes

**0**answers

114 views

### Are isolated knots in tangles the same as ordinary knots?

Suppose you have an arc in a ball with ends fixed on a boundary. In other words, we have an isolated knot on one of the arcs of a tangle. If we glue the ends we get an ordinary knot and clearly ...

**8**

votes

**2**answers

392 views

### Khovanov-Rozansky homology and spectral sequences

In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and ...

**3**

votes

**1**answer

149 views

### One question about the quandle

Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston ...

**8**

votes

**1**answer

238 views

### Existence of different knots in $RP^3$ having the equivalent liftings in $S^3$

I'm looking for the answer to following question. Do exist different knots in $RP^3$ which have equivalent liftings in $S^3$ under covering $p:S^3\rightarrow RP^3$?

**5**

votes

**1**answer

203 views

### Differences between various categories of surface embeddings in 4-space

This is a very naive question, but I'm trying to understand the difference between the various categories when it comes to embedding surfaces in 4-dimensional manifolds. The situation I'd really like ...

**3**

votes

**1**answer

122 views

### Some questions about ideal knots

The ropelength of a knot curve $C$ is defined as the ratio $L(C) = \lambda(C)/ \tau(C)$, where $\lambda(C)$ is the length of $C$ and $\tau(C)$ is the thickness of the knot defined by $C$ [from ...

**6**

votes

**2**answers

294 views

### Vassilliev invariants of knots and their cables

The following is perhaps a standard question, but i could not find a plain enough answer
by simply searching online.
Q: Given a knot $K$ and its $(p,q)$-cable $K_{p,q}$ what is a relation
between ...

**6**

votes

**1**answer

326 views

### integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...

**9**

votes

**3**answers

350 views

### Is a knotted trivalent graph determined by its set of unzips?

A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph in $\mathbb{R}^3$, modulo ambient isotopy. I believe they were first considered (in the quantum topological context, at ...

**7**

votes

**2**answers

337 views

### Original proof of the existence of Seifert surfaces

I read on Wikipedia that Frankl and Prontrjagin were the first to prove that a link $\mathbb{R}^3$ bounds a surface. A few years later Seifert published a proof using the "Seifert algorithm" which ...

**14**

votes

**4**answers

831 views

### Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question was originally posted on math.SE by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was made in ...

**6**

votes

**2**answers

289 views

### Hyperbolic Brunnian links and rectangular cusp shapes

My question is as follows.
Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?
Here is what I mean:
The Borromean rings form a famous link $B$ (a smooth ...

**1**

vote

**1**answer

202 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

252 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

179 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

221 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

230 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

482 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

278 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

159 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

246 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 ...

**5**

votes

**1**answer

649 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

206 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

324 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

220 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

152 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 ...

**33**

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

336 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

532 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 ...

**7**

votes

**1**answer

262 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

177 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

337 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

462 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

228 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 ...

**7**

votes

**4**answers

540 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

802 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

279 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. ...

**8**

votes

**0**answers

493 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

457 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

297 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

254 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

318 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 ...