Branched Spherical CR structures on the complement of the figure eight knot

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We obtain a branched spherical CR structure on the complement of the figure eight knot with a given holonomy representation (called rho₂). There are essentially two boundary unipotent representations from the complement of the figure eight knot into PU(2,1), we call them rho₁ and rho₂. 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 behaviour of of the fundamental group of the fiber under the representation is a key difference between rho₁ and rho₂.