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

TL;DR

This paper constructs a 1-parameter family of spherical CR uniformizations of the figure eight knot complement, showing how small deformations of the holonomy group yield new uniformizations with boundary holonomy remaining parabolic.

Contribution

It introduces an explicit 1-parameter family of deformations of the boundary unipotent holonomy group for the figure eight knot complement, expanding the understanding of its spherical CR uniformizations.

Findings

Existence of a simple fundamental domain for the holonomy group.

Small deformations preserve the uniformization with boundary holonomy parabolic.

Construction of a continuous family of non-conjugate uniformizations.

Abstract

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