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We obtain a branched spherical CR structure on the complement of the figureeight knot with a given holonomy representation (called rho_2). There areessentially two boundary unipotent representations from the complement of thefigure eight knot into PU(2,1), we call them rho_1 and rho_2. We make explicitsome 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 representationis a key difference between rho_1 and rho_2.

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