34
votes

### Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The answer of Jørgen Ellegaard Andersen only concerns the case of when the gauge group is $SU(n)$.
I will argue that all the ingredients for the equivalence between the two approaches (namely "...

25
votes

Accepted

### The Jones polynomial at specific values of $t$

The evaluation of the Jones polynomial at $e^{i\pi/3}$ is related to the number of 3-colourings $tri(K)$ of $K$ (see also here) as well as to the topology of the branched double cover $\Sigma(K)$:
$$...

15
votes

### Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

The simplest answer is that the universal cover together with the deck group action contain a lot of information about the manifold, and the representations of the group provide one way to ...

14
votes

### Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Three manifold groups are quite special and their representation varieties have more structure than that of a random group. Casson's view point is helpful to see the point. The Heegaard decompoistion ...

12
votes

Accepted

### Generators and relations for the 2-dimensional unoriented cobordism category

My initial answer was wrong, here's the correct version plus a reference: Turaev-Turner
New generating morphisms: The Mobius strip $\emptyset \rightarrow S^1$ and the "orientation reversing" ...

12
votes

Accepted

### Classification of unitary modular tensor categories (UMTCs)

Recently, Mignard and Schaunberg found a counter example to the conjecture (https://arxiv.org/abs/1708.02796). Counting $(S,T)$ pairs is still just a shorthand way of counting monoidal equivalence ...

12
votes

### Inverse Kirby knot

The standard model for the knot $K''$ you describe is simply the meridian $\mu_K$ of the knot $K$. To be a bit pedantic, this really means that you draw the meridian in the complement of $K$, and then ...

11
votes

### Proving that the Jones polynomial is q-holonomic

One statement that would imply that the colored Jones polynomials are q-holonomic involves the Kauffman bracket skein module $S_q(K)$ of the knot complement. This is a module over the skein module of ...

11
votes

### Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Here is another (very recent) motivation which might be more specific.
First, one application of the studying the $SL(2,\mathbb{C})$ character variety is that it detects some essential surfaces (i.e....

10
votes

### Hyperbolic Volume and Chern-Simons

The first reference known to me is
Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.
...

10
votes

Accepted

### Why are Witten-Reshetikhin-Turaev invariants expected to be integral?

My humble point of view is that the Witten-Reshetikin-Turaev invariant (at least for $G=SU(2)$ or $SO(3)$) at the root $\xi$ is (by definition) a rational function on $\xi$ which looks very much like ...

10
votes

### Lagrangian of Reshetikhin-Turaev TFT's

I don't think there's a way to extract a Lagrangian from the Reshetikhin-Turaev construction. There's certainly not
a unique way to do so.
Physicists believe that most QFTs are "non-Lagrangian,&...

8
votes

### Algebraic proof of 4-colour theorem?

There is an algebraic method by Alon and Tarsi which allows in certain cases to prove that certain graphs are $k$-colorable (in fact, even $k$-choosable). A famous case where this method prevails is ...

8
votes

Accepted

### Brauer-Picard for a fusion category coming from a quantum group

As far as I know, no one has written this up, but I think you should be able to find the Brauer-Picard groupoid for quantum groups at roots of unity by the following techniques. Now that I've written ...

8
votes

Accepted

### Hyperbolic Volume and Chern-Simons

This is a Theorem of Yoshida, the reference is
Yoshida, Tomoyoshi: ''The η-invariant of hyperbolic 3-manifolds.'' Invent. Math. 81, 473-514 (1985). http://mathlab.snu.ac.kr/~top/articles/Yoshida.pdf ...

8
votes

### P-adic Volume Conjecture

I don't know how to answer your question, since I don't know about motives or $p$-adic regulators (a reference would be helpful). I'll just point out one possible relation which may just be a ...

8
votes

### Lagrangian of Reshetikhin-Turaev TFT's

It's an open conjecture by Moore and Seiberg (originally in the context of conformal field theory) that every MTC can be obtained from Chern-Simons theory of simple Lie groups with known constructions....

7
votes

### Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

I would argue, as I do in the introduction of many of my papers, that the study of Betti Moduli Spaces $$\mathrm{Hom}(\Gamma, G)//G$$ for finitely generated $\Gamma$ and complex reductive $G$ is ...

7
votes

Accepted

### Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds

If your topological field theory is at least once-extended, by which I mean it assigns values to $(n+1)$-manifolds, $n$-manifolds, and also to $(n-1)$-manifolds, than this cannot happen.
More ...

7
votes

Accepted

### Polynomial invariants for unoriented links

Colored Kauffman polynomials are independent of orientations of links. If you look at the skein relation of Kauffman polynomial, there is no arrow. This is true for colored cases. Kauffman polynomials ...

7
votes

### The Jones polynomial at specific values of $t$

The volume conjecture predicts the existence limit of (a certain normalization of) the colored Jones polynomials evaluated at roots of unity (which is not known to exist), and that this limit is equal ...

7
votes

Accepted

### Non-extendable 3D TQFTs

Check out Andras Juhasz' paper: https://arxiv.org/pdf/1408.0668.pdf
Specifically, Theorem 1.10:
There is an equivalence between the symmetric monoidal category
of (2+1)-dimensional TQFTs and the ...

6
votes

### Algebraic proof of 4-colour theorem?

There are several reformulations by Matiyasevich, like this with polynomial, this with binomial coefficients modulo 7, also some others but they are less `algebraic', whatever it means.

6
votes

### Is there a volume conjecture for closed 3-manifolds?

There's another volume conjecture formulated by Chen and Yang for Turaev-Viro invariants of closed manifolds. They present some evidence for the conjecture in the paper. In a second paper, Yang and ...

5
votes

### Polynomial invariants for unoriented links

Let $\mathfrak g$ be a simple Lie algebra, $U_q(\mathfrak g)$ the corresponding quantum group and $V$ a simple module over it. From this you get an invariant of framed oriented links. Now since $V$ is ...

5
votes

### Jones polynomial of tangles using Temperley-Lieb algbra?

Yes, the same thing can be done using in terms of the TL algebra. Namely, that is how the Jones polynomial was originally defined. For TL, there are two ways to get the link invariant, which both ...

5
votes

Accepted

### Does the limit in the Volume conjecture converge?

No, this is unknown. There are heuristic arguments for convergence based on the stationary phase approximation, but as far as I know, no one has made the argument precise in general. The closest I ...

4
votes

### Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

For future reference. The elegant argument sketched by Kevin Walker above involving pants decompositions and Bohr-Sommerfeld fibers was published by Andersen on the arXiv later that year, as part of ...

4
votes

### Relations between quantum groups at roots of unity, modular representation theory, and physics

Modular representations (representations in spaces over a field of nonzero characteristics) have been used in physics by Felix Lev to construct a quantum theory that is based on a finite number field (...

3
votes

Accepted

### Quantum homology of $(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$ and Poincare duality

A bit late for this one, but I'll still post the answer for future visitors.
Poincaré duality on the quantum homology is just the same as Poincaré duality on normal homology, see for example the ...

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