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We describe a simple fundamental domain for the holonomy group of theboundary unipotent spherical CR uniformization of the figure eight knotcomplement, and deduce that small deformations of that holonomy group (suchthat the boundary holonomy remains parabolic) also give a uniformization of thefigure eight knot complement. Finally, we construct an explicit 1-parameterfamily of deformations of the boundary unipotent holonomy group such that theboundary holonomy is twist-parabolic. For small values of the twist of theseparabolic elements, this produces a 1-parameter family of pairwisenon-conjugate spherical CR uniformizations of the figure eight knot complement.

A 1-parameter family of spherical CR uniformizations of the figure eight knot complement