# Tagged Questions

**11**

votes

**1**answer

591 views

### Kontsevich integral : state of the art

The Kontsevich integral is known to be a universal Vassiliev invariant.
It is still an open question whether it is a complete knot invariant, i.e. whether it distinguishes a given knot from all other ...

**3**

votes

**0**answers

123 views

### Is a generic link diagram semi-adequate?

Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution.
Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a ...

**7**

votes

**1**answer

394 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 ...

**8**

votes

**3**answers

340 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 ...

**3**

votes

**0**answers

264 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. ...

**5**

votes

**1**answer

281 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, ...

**12**

votes

**1**answer

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

**4**

votes

**1**answer

514 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:
...

**7**

votes

**2**answers

584 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 ...

**4**

votes

**1**answer

211 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

**1**answer

491 views

### Recovering the Alexander Polynomial from Ocneanu's HOMFLYPT

Let $H_q(n)$ denote the Hecke algebra associated to the symmetric group $S(n)$: this is the $\mathbb{Z}[q^{\pm 1/2}]$ algebra generated by $T_1, \ldots, T_{n-1}$ satisfying the braid relations along ...

**8**

votes

**2**answers

633 views

### Torus knots in Euclidean space — a symmetry argument

Consider a $(p,q)$ torus knot $K$ in 3-dimensional Euclidean space $\mathbb R^3$ where $p,q \geq 2$ and $\operatorname{GCD}(p,q)=1$.
Let $\operatorname{Isom}(\mathbb R^3,K)$ be the isometries of ...

**2**

votes

**1**answer

183 views

### Invariants of unframed, oriented links from Reshetikhin Turaev construction

Hello!
I have a few questions on Reshetikhin Turaev invariants.
By RT any ribbon category ${\mathcal C}$ yields an invariant of oriented, framed links labelled with objects of ${\mathcal C}$.
Is ...

**2**

votes

**1**answer

446 views

### Now that I got a mutant-discriminating invariant…

...what can I do with the darn thing?
Background: I read that still no Vassiliev Invariant with mutant-discriminating
power is known (correct me if this is outdated). Now, my research lead to a
whole ...

**0**

votes

**1**answer

324 views

### Some Vassiliev Invariant Questions

The V.I. definition goes doublepoint=overpass-underpass (or was it the other
way around? If it's 50:50, I score 0 always :-). Would it lead anywhere
to define doublepoint=overpass+underpass? (Even if ...

**8**

votes

**0**answers

277 views

### Which presentations of (non)planar algebras give rise to knots?

Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...

**16**

votes

**1**answer

1k views

### How are the Conway polynomial and the Alexander polynomial different?

Background story:
I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he ...

**1**

vote

**3**answers

734 views

### SO(3) knot polynomials

Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix ...

**17**

votes

**2**answers

1k views

### Why is the Alexander polynomial a quantum invariant?

When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander ...