13
votes
SL(2, C)-representation of a knot
$(P)SL(2, \mathbb{C})$ is the isometry group of $\mathbb{H}^3,$ so $SL(2, \mathbb{C})$ representations are the natural generalization of hyperbolic structures on knot complements.There is a vast ...
- 96.4k
11
votes
Moduli space of flat connections of Lie group over a 2-torus
Let $K$ be a connected compact Lie group. The moduli space of flat $K$-bundles over an $n$-torus is homeomorphic to the character variety $Hom(\mathbb{Z}^n,K)/K$.
The identity component of this ...
- 8,529
9
votes
SL(2, C)-representation of a knot
The fundamental group of the complement of a knot in $S^3$, called a knot group, is a knot invariant (equivalent knots have the same knot group, but not conversely). To understand knot groups, ...
- 8,529
8
votes
Character variety of the free group
I think it depends on what you want.
The $\mathrm{SL}_2(\mathbb{C})$-character variety of a free group $F_r$ admits a model over $\mathbb{Z}$ (maybe its better to say $\mathbb{Z}[1/2]$); see Rank 1 ...
- 8,529
8
votes
Accepted
Hyperbolic homology spheres with infinite $\mathrm{SL}_2(\mathbb{C})$ character variety
Take two knot complements (say of $K,K'$) and glue them together, interchanging meridians and longitudes. This is called splicing and produces a homology sphere $S(K,K')$; if both knots are non-...
- 19.4k
7
votes
Accepted
Why is the $\operatorname{GL}_n$ character variety "cohomologically" the product of the $\operatorname{PGL}_n$ character variety and a torus?
You need to keep reading to the proof of Theorem 2.2.12.
The main point is that the $PGL_n$-character variety $\tilde{\mathcal{M}}_n/\mathbb{C}$ is both a quotient by a torus $\mathcal{M}_1/\mathbb{C}\...
- 8,529
5
votes
Accepted
Character variety of the free group
Let $\pi$ be a free group of rank 2. The character variety of $SL_{2,k}$-representations of $\pi$ is always isomorphic to affine 3 space $\mathbb{A}^3_k$ for any ring $k$.
Let $A[\pi] = A[\pi]_k$ ...
- 10.7k
5
votes
Accepted
Distinct knots with same $A$-polynomial
The torus knots $T_{7,15}$ and $T_{3,35}$ have the same A-polynomials. In general, if $p,q>1$ are coprime and odd then $T_{p,q}$ has A-polynomial $(L-1)(LM^{pq}+1)(LM^{pq}-1)$, which only depends ...
- 6,579
5
votes
Accepted
What is the state in the WRT TQFT associated to a handlebody?
I can think of two cases where the Witten-Reshetikhin-Turaev vector $Z_k(Y^3)\in Z_k(\Sigma)$ has been connected to a Lagrangian state as you described:
-Laurent Charles and Julien Marché showed that ...
- 492
5
votes
Accepted
Moduli space of semistable bundles
Disclaimer: This answer is rewritten in response to Chris Woodward's insightful comments.
The moduli space of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 ...
- 8,529
4
votes
Compactifications of SL(2)-character varieties of surfaces
I don't think this actually answers any of your questions explicitly, but it might help.
In my recent paper Wonderful Compactification of Character Varieties (co-authored with I. Biswas and D. Ramras)...
- 8,529
4
votes
SL(2, C)-representation of a knot
This is a good introduction:
Shalen, Representations of 3-manifold groups. Handbook of geometric topology, 955–1044, North-Holland, Amsterdam, 2002.
- 10.6k
4
votes
Accepted
Is a local system on a surface determined by simple closed loops?
$\def\ZZ{\mathbb{Z}}\def\FF{\mathbb{F}}$This is an attempt to fix a (currently deleted) answer of Will Sawin's. Will tried to show that the claim was false for $n = 3^g$. I can show, by a similar ...
- 156k
4
votes
Is a local system on a surface determined by simple closed loops?
The n=2 case is implied by the proof of Theorem 2.1 in:
https://arxiv.org/abs/0901.1402
For the one-holed torus and n=3, the result is in my thesis.
- 8,529
4
votes
Accepted
Orbit of an irreducible representation of a surface group under that action of the mapping class group
Here is a partial answer (showing there exists a Zariski dense collection of irreducible representations with the requisite property).
In Topological Dynamics on Moduli Spaces II by Previte, and Xia ...
- 8,529
3
votes
Accepted
Lie bracket on the complex valued functions of the space of representations of a Riemann surface
In the complex case, Goldman's symplectic form is a $(2,0)$-form, and the Poisson bracket is in terms of that form (the bracket is determined by a bivector and the bivector corresponds to the form).
...
- 8,529
3
votes
Distinct knots with same $A$-polynomial
I quote (page 753) from:
Cooper, D., Long, D. D., Representation theory and the A-polynomial of a knot., Knot theory and its applications. Chaos Solitons Fractals 9 (1998), no. 4-5, 749-763.
"It ...
- 8,529
3
votes
Is the irreducible locus of the character variety a principal bundle in Zariski topology?
First, let's assume that the genus $g$ of $\Sigma$ is greater than or equal to 2 (otherwise the irreducible locus might be empty if $G$ is non-abelian).
Then for most choices of $G$, the answer is no, ...
- 8,529
3
votes
Orbit of an irreducible representation of a surface group under that action of the mapping class group
As some further evidence for a positive answer to your question,
there is a paper of Cantat and Loray that proves a relative version of this for mapping class group the 4-holed sphere (rel boundary). ...
- 68.9k
3
votes
A question about dimension of SL(2,C) character variety of knot group
In Incompressible surfaces in tunnel number one knot complements by Mario Eudave-Muñoz, Figure 7 gives an example of a knot with a closed essential surface. The author guesses it is the least volume ...
- 8,529
3
votes
Classification of geometric structures through character varieties
This question was pretty much answered in the comments, so I will just give some relevant references:
Geometric structures on manifolds and varieties of representations, by W. Goldman, Geometry of ...
- 8,529
3
votes
The ¨irreducible¨ representation variety of surface group
Let $\Sigma$ be a closed orientable surface of genus $g\geq 2$, and $\pi$ its fundamental group. Let $X(\pi, G)=Hom(\pi,G)/G$ be the conjugation quotient for $G$ a compact Lie group whose derived ...
- 8,529
3
votes
Kernel of restriction for ring of functions on reductive groups
Even in the case $n=1$, the map $k[G^n]^G\to k[H^n]^H$ is not injective.
Example: let $G=GL_2(\mathbb{C})$ and $H=SL_2(\mathbb{C})$. Then $\mathbb{C}[H]^H\cong\mathbb{C}[t]$ where $t$ corresponds ...
- 8,529
2
votes
Accepted
Metric on moduli space of semistable principal G-bundles on curves
Let $G$ is a reductive linear algebraic group over $\mathbb{C}$, and $X$ be a connected compact Riemann surface of genus $\geq 2$.
Fix a topological type $\tau$. Then the moduli space of semistable ...
- 8,529
1
vote
Orbit closures in smooth irreducible components
Let $V$ be an irreducible representation of a reductive group $H$ such that every orbit has codimension $\ge2$ (e.g., when $\dim V\ge\dim H+2$). Put $G:=\mathbb C^*H$ with $\mathbb C^*$ acting by ...
- 15.5k
1
vote
Reference on representations of knot groups
Representations on dihedral groups and the symmetric group S4 are explained in my 1975 and 1976 papers: "Octahedral knot covers" and "On dihedral covering spaces of knots." They are easy to see, and ...
- 23
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