5
votes
1answer
220 views

Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc. My question is ...
10
votes
1answer
246 views

Representation varieties of 3-manifold groups in SL(n,C)

I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $SL(n,C)$: $$Hom(\pi_1M, SL(n,{\mathbb C}))$$ It is known that volume and Chern-Simons ...
4
votes
2answers
332 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, ...
3
votes
0answers
283 views

$ABA^{-1}B^{-1} = E$ the topology of the space of non-commuting matrices

Is there any discussion of topology of space of matrices $ ABA^{-1}B^{-1} = E $ with $E$ diagonal matrix, $A,B \in \mathrm{GL}(n,\mathbb{C})$? E.g. is this a variety of just a scheme? How many ...
10
votes
3answers
359 views

Topology on the Unitary Dual

Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of ...
4
votes
2answers
269 views

Complexification or 'real'ization of Mapping Class group.

So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
6
votes
2answers
235 views

equivariant cohomology of the complement to the arrangment $\cup_{i\neq j}overrightarrow{x_i} = overrightarrow{x_j}$?

Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidian) vector space over real numbers. Let $G=SO(V)$ be a compact Lie group of linear orthogonal transformations of $V$. Let $Conf_n(V)$ be the space of ...
8
votes
0answers
311 views

What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds ?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
7
votes
1answer
167 views

Algorithmic Borel Finiteness?

It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of ...
3
votes
1answer
239 views

Kernel of the representation of the mapping class group to $Aut(F_n)$

Let $S_{g,1}$ be a orientable compact surface of genus $g$ with one boundary component and $\Gamma_{g,1}$ the mapping class group. By $F_n$ I denote the free group on $n$ generators. One obtains a ...
8
votes
2answers
380 views

Cohomology of representation varieties

Perhaps this question is too general then I am sorry about this. My question is the following. Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) ...
15
votes
7answers
3k views

Topology of SU(3)

$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...
6
votes
1answer
386 views

What is the image of the half/full twist in the Hecke algebra, in the Kazhdan-Lusztig basis? What is the corresponding complex of Soergel bimodules?

Let $B_n$ be the braid group on $n$ strands. It has generators $\tau_i$ for $i = 1,\ldots,n-1$ which exchange the $i$th and $(i+1)$st strands, and which satisfy the relations $\tau_i \tau_j = \tau_j ...
17
votes
2answers
568 views

Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants

Recall that there are knots in $\mathbf{R}^3$ that are not invertible, i.e. not isotopic to themselves with the orientation reversed. However, it is not easy to tell whether or not a given knot is ...
1
vote
2answers
418 views

Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...