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 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 $X_n(M)$ ?
 A: 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:

*

*Michael Heusener

*Emily Landes

*Melissa Macasieb

*Vicente Muñoz

*Kate Petersen

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

*

*Geometry of the SL(3,C)-character variety of torus knots, by
Vicente Muñoz, Joan Porti for torus knots and $n=3$, and

*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:

*

*Local coordinates for SL(n,C) character varieties of finite volume hyperbolic 3-manifolds, by Pere Menal-Ferrer, Joan Porti and,

*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).
