All Questions
Tagged with qa.quantum-algebra knot-theory
56 questions
1
vote
0
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125
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Tangle hypothesis and ribbon category
The tangle hypothesis, when specialized to ordinary framed tangles, says that the framed tangles form the free braided category with all duals (i.e. considered as a 3-category, all the 1- and 2-...
- 1,263
4
votes
0
answers
166
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Coloured Jones polynomial at 4th root of unity and Arf invariant
Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
- 81
13
votes
2
answers
1k
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 $...
- 4,713
8
votes
0
answers
233
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$U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$.
Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
- 12.9k
24
votes
3
answers
3k
views
How many definitions are there of the Jones polynomial?
Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
- 68.9k
9
votes
1
answer
297
views
Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?
For an (oriented) knot in $S^3$ the number $\Gamma(K) := \Delta_K’’(1)$ shows up in a number of places in knot theory, for example the Casson-Walker-Lescop invariant. Here $\Delta_K(t)$ is the ...
- 44.4k
5
votes
1
answer
525
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Motivating quantum groups from knot invariants
Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...
- 3,357
28
votes
2
answers
3k
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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 ...
- 4,713
3
votes
1
answer
374
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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)
\...
- 4,713
3
votes
0
answers
134
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Relative strength of Jones and colored Jones polynomials
this is my first post here.
I've been studying some Knot Theory and I came to a question concerning invariants.
We know that the Jones polynomial is related to the RT-invariant associated to the two-...
- 63.9k
19
votes
4
answers
2k
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What are the points of Spec(Vassiliev Invariants)?
Background
Recall that a (oriented) knot is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of ...
- 35.5k
4
votes
1
answer
168
views
Link invariants from modular categories (strictification and computation)
By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
- 28.1k
1
vote
0
answers
111
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Rack cohomology as derived functor cohomology
Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...
- 1,066
3
votes
1
answer
188
views
What is the determinant of the R-matrix defining the colored Jones polynomial?
Let $V_n$ be the $(n+1)$-dimensional irreducible representation of $\mathcal U = \mathcal{U}_q(\mathfrak{sl}_2)$, and let $\mathbf R \in \mathcal{U} \widehat \otimes \mathcal{U}$ be the universal $R$-...
- 2,533
2
votes
0
answers
292
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Tracking down an elusive book
A few weeks ago I had a very engaging talk with a faculty member, where he told me lots of interesting things about quantum algebras, know theory and Reshetikhin-Turaev invariants (this field is not ...
- 697
10
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1
answer
324
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Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?
Both the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar ...
- 2,533
10
votes
2
answers
664
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 ...
- 630
12
votes
2
answers
789
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 ...
- 2,533
7
votes
0
answers
209
views
IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists
Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc.
Let $G_{g,b}$ denote the set of finite ...
- 40.2k
6
votes
0
answers
156
views
Relation between different versions of Bar-Natan homology
In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by ...
- 13.6k
4
votes
1
answer
265
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Framing dependence of HOMFLY polynomial
I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...
- 1,430
6
votes
1
answer
302
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Knot Factorization Homology inputs
Following the paper by Ayala, Francis, and Tanaka: https://arxiv.org/pdf/1409.0848.pdf
If we are talking about knots we are talking about framed 3-manifolds with a framed 1-dimensional sub-manifold ...
- 9,481
5
votes
1
answer
348
views
Presentations of the monoidal categories of virtual tangles and of welded tangles by generators and relations
Reidemeister theorem implies, without too much fuss, that the monoidal categories of tangles, and of oriented tangles, can be presented by generators and relations. This is done for example in
a) ...
- 8,524
8
votes
0
answers
597
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 $0\leq k\leq n/2$.
\begin{align}
P_{0,0}&=1\\
\text{for $n\geq 1$}...
- 19.3k
7
votes
1
answer
986
views
Understanding "Decategorified" symplectic Khovanov homology
In http://arxiv.org/abs/math/0405089 Seidel and Smith constructed a link invariant using Lagrangian Floer theory that was conjectured to be equivalent to Khovanov homology. The equivalence was ...
CommunityBot
- 1
4
votes
1
answer
278
views
Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?
Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by $...
CommunityBot
- 1
21
votes
1
answer
863
views
Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?
The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...
- 1,129
4
votes
1
answer
959
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Jones polynomial of tangles using Temperley-Lieb algbra?
The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...
- 2,121
8
votes
1
answer
413
views
Does the limit in the Volume conjecture converge?
The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...
- 68.9k
0
votes
0
answers
373
views
Understanding a program for computing Khovanov homology
I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp
The ...
- 803
6
votes
1
answer
740
views
Kauffman's state model for the Alexander polynomial, via representation theory
I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...
CommunityBot
- 1
5
votes
2
answers
554
views
Jones polynomial of the concatenation of two braids
Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings.
Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$?
Here, $J_L(q)...
- 1,329
6
votes
0
answers
196
views
Software for BMW algebra calculations?
Does software exist for computations in the BMW algebra?
For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...
- 6,210
10
votes
1
answer
410
views
Links which HOMFLY homology distinguish but the HOMFLY polynomial does not.
Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links?
I'm assuming there does exist such ...
- 2,273
5
votes
3
answers
1k
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 ...
CommunityBot
- 1
1
vote
1
answer
220
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 ...
- 28.1k
5
votes
0
answers
346
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. ...
- 22.1k
6
votes
1
answer
1k
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 ...
CommunityBot
- 1
8
votes
2
answers
1k
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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 ...
- 3,474
5
votes
1
answer
340
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 ...
- 28.1k
7
votes
2
answers
693
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 ...
- 4,829
8
votes
1
answer
1k
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:
$\frac{1}...
- 2,869
1
vote
0
answers
165
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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 ...
- 4,829
6
votes
1
answer
396
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). ...
- 12.8k
4
votes
0
answers
300
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 ...
- 8,524
1
vote
1
answer
402
views
Knot polynomials: Skein>Matrix>Group?
OK, the heading was a bit tersely formulated...
If you have a quantum group and an irrep, you theoretically know the
R matrix (mathematicians are a notoriously idle lot, they give the
general formula ...
- 1,639
4
votes
1
answer
697
views
How does one relate the monodromy of the KZ equations with the WRT representation of the braid group?
The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll ...
- 1,639
7
votes
0
answers
363
views
Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?
Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, $...
- 5,264
6
votes
1
answer
921
views
Is the complete functorial structure for Khovanov--Lee homology known?
I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$.
The groups $\operatorname{Kh}_{\...
- 12.8k
18
votes
4
answers
2k
views
Who thought that the Alexander polynomial was the only knot invariant of its kind?
I apologize that this is vague, but I'm trying to understand a little bit of the historical context in which the zoo of quantum invariants emerged.
For some reason, I have in my head the folklore:
...
- 18.7k