Skip to main content
5 of 6
Updated links since the "front" is no longer active.
Sean Lawton
  • 8.5k
  • 3
  • 46
  • 78

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).

Sean Lawton
  • 8.5k
  • 3
  • 46
  • 78