Branched spherical CR structures on the complement of the figure-eight knot

Abstract. We obtain a branched spherical CR structure on the com- plement of the figure-eight knot whose holonomy representation was given in [4]. There are essentially two boundary unipotent representations from the complement of the figure-eight knot into PU(2,1), which we call ρ1 and ρ2. We make explicit some fundamental differences between these two representations. For instance, seeing the figure-eight knot complement as a surface bundle over the circle, the behavior of the fundamental group of the fiber under the representa- tion is a key difference between ρ1 and ρ2.