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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 ...
Liviu Nicolaescu's user avatar
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 ...
Tom Mrowka's user avatar
  • 3,014
14 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 ...
Matthew Titsworth's user avatar
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 ...
Danny Ruberman's user avatar
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" ...
Noah Snyder's user avatar
  • 27.8k
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....
Neil Hoffman's user avatar
  • 5,231
10 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 ...
ThiKu's user avatar
  • 10.3k
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. ...
Igor Rivin's user avatar
  • 95.6k
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,&...
Arun Debray's user avatar
  • 6,766
9 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....
Meng Cheng's user avatar
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 ...
Ian Agol's user avatar
  • 66.9k
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 ...
André Henriques's user avatar
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 ...
Sean Lawton's user avatar
  • 8,394
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 ...
Peter Samuelson's user avatar
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 ...
Ian Agol's user avatar
  • 66.9k
5 votes
Accepted

Squier's conjecture on Burau at roots of unity

Presumably, this conjecture appears from the observation that the suitable powers of the standard generator $\sigma_1$ lies in the kernel, although it looks that Squier missed other obvious elements ...
Tetsuya Ito's user avatar
4 votes
Accepted

Does $U_q (\mathfrak{sl}_2)$ have a universal $R$-matrix?

$U_q(\mathfrak{sl}_2)$ is indeed not quasitriangular with the usual $R$-matrix. Instead it satisfies a weaker condition of being braided [1]. One can certainly show that the usual universal R-matrix ...
Calvin McPhail-Snyder's user avatar
4 votes

Is there an "arithmetic cobordism category"?

Unfortunately, I think we're still quite far away from knowing the answer to your question. Fortunately, a lot of progress has been made without answering your question. The key advance here was ...
NZK's user avatar
  • 345
4 votes
Accepted

Applications of quantum representations of the mapping class group to quantum computers

The context is topological quantum computation, where quantum information is stored nonlocally in a physical system, so that it is protected from decoherence by local sources of noise. The nonlocal ...
Carlo Beenakker's user avatar
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 (...
Carlo Beenakker's user avatar
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 ...
bas's user avatar
  • 186
3 votes

Computation of \tau invariant

See Livingston's "Computations of the Ozsvath-Szabo knot concordance invariant" (https://arxiv.org/abs/math/0311036), Corollary 3. Or see my thesis for a picture of Livingston's cobordism (p. 19, ...
Lukas Lewark's user avatar
2 votes

S-matrix for the HOMFLY/Hecke category

This isn't a complete answer, but it might make some partial progress. First I'll slightly restate the question. If we decompose $S^3$ into two solid tori and write $C$ and $C^{op}$ for the Homfly ...
Peter Samuelson's user avatar
2 votes

Does WRT invariant detect hyperelliptic involution on the genus 2 surface?

Unless I'm misunderstanding what's meant by hyperelliptic involution, WRT invariants do, in fact, detect the hyperelliptic involution in genus 1. Let $C$ be the modular tensor category (MTC) ...
Kevin Walker's user avatar
  • 12.4k
2 votes

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

The area seems to be very broad, let me just give some remarks, which are somewhat close to me. Many interesting conformal field theories are "rational", in some simple cases it means that some ...
Alexander Chervov's user avatar
2 votes
Accepted

Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?

First of all, however your universal finite type invariant $Z$ is given the question of computing the coefficient of a given diagram is somewhat ill-defined, since those diagrams are not linearly ...
Adrien's user avatar
  • 8,369
1 vote

Link invariants from modular categories (strictification and computation)

Yes, if you have equivalent ribbon categories then they give the same link invariants. This is just saying you can define the invariant in the language of ribbon categories. A helpful analogy is why ...
Noah Snyder's user avatar
  • 27.8k
1 vote
Accepted

How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?

Based on the discussion in the comments with Ian Agol, here's a draft answer. I would welcome corrections/confirmation from anyone who knows more. Let $M$ be an orientable manifold with possibly ...
Calvin McPhail-Snyder's user avatar
1 vote
Accepted

Framing dependence of HOMFLY polynomial

It turned out that it was a very simple calculation to compute $\langle \square,\square+2\rho\rangle$ for $sl(n)$ using Cartan matrices. For $sl(n)$, $\langle \square,\square+2\rho\rangle = 2-\frac{1}{...
Henry's user avatar
  • 1,420

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