Below you can get an idea of how Limit sets deform by changing a representation inside the character variety of the 8-knot complement with target group PU(2,1). Here, each orange point correspond to a (computed) representation of the fundamental group of the 8-knot complement in SU(2,1), as parametrized in Falbel-Guilloux-Koseleff-Rouillier-Thistlethwaite.
Some information from the literature:
What to do: By hovering above an orange point, you get below 3 snapshots of the limit set of the associated representation. Indeed, these limit sets lie in a 3-dimensional sphere(the boundary at infinity of the complex hyperbolic plane). They are projected to the space R3 by a stereographic projection. The 3 snapshots are then taken along each axis.
Warning! This implementation is rather slow. Be carefull not to move your mouse too fast. You may have to wait a few seconds for image loading.For more convenient exploration, you may want a local version of this page: just download this archive (~250Mo), unzip, and open the file LandscapeLimitSet.html in your browser.