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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 $X_n(M)$ ?

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There's this paper: – Ian Agol Feb 11 '12 at 22:09
and these: and many, many more, as well as many articles on SU(n) representations for n>2. Follow the references on these. – Paul Feb 12 '12 at 3:57

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 here for torus knots and $n=3$ and here for the figure eight knot and $n=3$.

For your second question, I recommend reading about tangent spaces to character varieties here for generalities. With respect to local deformations for (finite volume hyperbolic) 3-manifold groups, this and this answers your second question positively.

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 here (and references therein).

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