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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: front.math.ucdavis.edu/1111.4338 –  Ian Agol Feb 11 '12 at 22:09
and these: arxiv.org/abs/0710.3511 arxiv.org/abs/0803.4329 arxiv.org/abs/0902.2589 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

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