A Landscape of Limit Sets Deformations

The visualisation on this page gives 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 point correspond to a (computed) representation of the fundamental group of the 8-knot complement in SU(2,1), as parametrized in Guilloux-Will. It is a group generated by two elements, a and b, of order 3, with the additional property that the product ab has order 4. The parameter u below defines the four parameter of Guilloux-Will. Namely, it sets: z1=z3=1, z2 = u and z4 is the conjugate of z2.

Some information from the literature:

0
0
u = 3
tile snapshot
Cartan map

There is still a lot of work to do to build what I would like for this landscape. This includes - but is not limited to - the following:

For most of these tasks, I am aware of a solution... but there are some difficulties (time, sheer size of the files, my poor JS abilities...).

By Raphael Alexandre, Vincent Cornet, Antonin Guilloux, Thi Thu Quyen Nguyen. August 2021.