All Questions
Tagged with qa.quantum-algebra quantum-topology
30 questions
3
votes
1
answer
165
views
quantum invariants, ribbon Tannakian duality and classification of ribbon Hopf algebras
In a nutshell, my question is:
Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction?
I will now make it more precise. One could define a ...
- 213
4
votes
0
answers
165
views
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
6
votes
1
answer
223
views
Does $U_q (\mathfrak{sl}_2)$ have a universal $R$-matrix?
Consider the standard quantum group $U_q (\mathfrak{sl}_2)$ over the field $\mathbb{C}(q)$ of rational functions (or over $\mathbb{C}$ if $q \in \mathbb{C}$ is not a root of unity), with the usual ...
- 601
3
votes
1
answer
144
views
Closed-form expressions for the Kashaev invariant via surgery
For a knot $K$, let $J_N(K)$ denote the $N$th Kashaev invariant of $K$. This is the same as the $N$th colored Jones polynomial evaluated at an $N$th root of unity (or $2N$th depending on your ...
- 2,533
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 ...
- 5,230
1
vote
0
answers
79
views
Ordering in Cobordism Category
Let $Cob^{3}$ denote the cobordism category of $1$ dimensional manifolds i.e the objects are finite disjoint union of circles and morphisms are represented by surfaces.
Is it possible to treat the ...
- 341
5
votes
1
answer
331
views
How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?
Chen and Yang have a more general version of the volume conjecture that they state for all hyperbolic $3$-manifolds (Conjecture 1.1 of [2]) including those with boundary. To do this, they have to ...
- 2,533
3
votes
0
answers
96
views
Are the tangle functors based off Khovanov homology braided monoidal functors?
I was wondering if the tangle functors constructed in
"A functor-valued invariant of tangles"
https://arxiv.org/pdf/math/0103190.pdf
"An invariant of tangle cobordisms via subquotients of arc rings"
...
- 91
6
votes
1
answer
217
views
Applications of quantum representations of the mapping class group to quantum computers
Quantum representations of the mapping class group of a surface are certain representations constructed from the data of a TQFT and described, for example, in and 1 and 2.
The following sources 3 ...
- 2,964
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
4
votes
1
answer
265
views
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
13
votes
1
answer
1k
views
Classification of unitary modular tensor categories (UMTCs)
Context/background:
I'm approaching this topic from the perspective of anyonic systems.
In the study of anyons, one works with fusion categories. Of course, for physicality, we demand that
i) The ...
- 318
4
votes
1
answer
959
views
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 ...
- 803
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
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...
- 2,869
13
votes
1
answer
848
views
Are Turaev--Viro invariants secretly a discretized path integral?
Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with $\left|WRT(M)\...
- 18.7k
8
votes
1
answer
510
views
Reference request: the "Kauffman bracket skein category"?
There should be a category $3\text{CobTang}$ whose
objects are some kind of surfaces with a finite set of marked points
morphisms $M : S \to T$ are some kind of $3$-dimensional cobordisms $...
- 118k
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. ...
- 2,273
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
67
votes
4
answers
9k
views
Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?
The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
- 18.7k
2
votes
1
answer
549
views
Why is the quantum Lorentz group not connected?
Podles and Woronowicz' construct the quantum Lorentz group, by which they mean $SL_q(2,\mathbb{C})$, as a quantum double of the compact quantum group $SU_q(2)$. More precisely, it is a bicrossed ...
- 853
15
votes
0
answers
1k
views
Representations of quantum groups at roots of unity
I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
- 18.7k
7
votes
1
answer
556
views
Quantum coordinate ring at root of unity
Noah Snyder gave a great answer to this question about different versions of a quantum group $U_q(\mathfrak g)$ when $q$ is a root of unity. I want to ask about forms of the deformed coordinate ring $...
- 18.7k
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 ...
- 141
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). ...
- 18.7k
3
votes
1
answer
569
views
What if I change field in a Topological Quantum Field Theory?
Of course I'm talking about the algebraic notion of field.
In a few words, if a TQFT consists of a functor $Z\colon Cob(n)\to \mathbf{Vec}_k$, I'm wondering if there are sensible relations among ...
- 13.6k
21
votes
3
answers
12k
views
Quantum mathematics?
"Quantum" as a term/prefix used to be genuinely physical: what was supposed to be physically continuous turned out to be physically quantized.
What sense does this distinction make inside ...
17
votes
1
answer
1k
views
Is there an "arithmetic cobordism category"?
This question is a clumsy attempt to apply a certain analogy. I hope that if the answer is negative it comes with a clarification of the scope and limitations of the analogy.
Arithmetic topology is ...
- 118k
1
vote
3
answers
995
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 ...
- 1,129
20
votes
4
answers
2k
views
Is there a volume conjecture for closed 3-manifolds?
A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is
Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...
- 3,093
28
votes
2
answers
3k
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 ...
- 22.1k