# 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 $X_n(M)$ ?

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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

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:

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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