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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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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).

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  • $\begingroup$ Hi Sean, the arXiv 'front' site is down indefinitely, would you please update the links that point to papers/authors there? Having a 'this' or 'here' as a link work is also not good for future-proofing, giving the titles of the papers at a minimum would be better. $\endgroup$
    – David Roberts
    Sep 21, 2021 at 1:14
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    $\begingroup$ @DavidRoberts I have updated the links as you requested, and added the titles (editing the answer accordingly). Thanks for your interest in my answer! $\endgroup$ Sep 22, 2021 at 2:39
  • $\begingroup$ Thanks heaps! there are lots of these links, some more obscure than others. I'm slowly chipping away at the ones that can't be mass-edited with some kind of script. $\endgroup$
    – David Roberts
    Sep 22, 2021 at 2:47

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