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

Question

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 known explicitly for torus knots, and tabulated for many hyperbolic knots, e.g. on Culler's webpage.

I am wondering if the $SL(n,{\bf C})$ character varieties (denote it $X_n(M)$) are also known (by a set of polynomial equations for instances) in some examples, like the complement of torus knots, or of the figure-8 knot ?

Another question: it is known that $X_n$ at a smooth point has complex dimension $(n - 1)$. Does there exist "remarkable" subvarieties of complex dimension $1$ in $X_n(M)$ ?

Answer 1

The answers at this related question might be of interest.

As implied by the comments, there is a large body of work on this topic.

Here are some authors (definitely not exhaustive) who have worked out the exact structure of character varieties of 3-manifold groups:

1. Michael Heusener
2. Emily Landes
3. Melissa Macasieb
4. Vicente Muñoz
5. Kate Petersen
6. Joan Porti

In particular, the answer to your first question is yes. See:

1. Geometry of the SL(3,C)-character variety of torus knots, by Vicente Muñoz, Joan Porti for torus knots and $n=3$, and
2. The SL(3,C)-character variety of the figure eight knot, by Michael Heusener, Vicente Munoz, Joan Porti for the figure eight knot and $n=3$.

For your second question, I recommend reading generalities about tangent spaces to character varieties in:

Character Varieties, by Adam Sikora.

With respect to local deformations for (finite volume hyperbolic) 3-manifold groups, the following references answers your second question positively:

1. Local coordinates for SL(n,C) character varieties of finite volume hyperbolic 3-manifolds, by Pere Menal-Ferrer, Joan Porti and,
2. Twisted cohomology for hyperbolic three manifolds, by Pere Menal-Ferrer Joan Porti

As to the third question, I am not sure what "remarkable" means here, so I will just leave that one alone.

Another interesting part of the story of character varieties of 3-manifold groups concerns dynamics. See the very nice exposition by Dick Canary titled Dynamics on character varieties: a survey (and references therein).