Complex hyperbolic geometry of the figure eight knot

TL;DR

This paper demonstrates that the figure eight knot complement can be endowed with a unique spherical CR structure derived from complex hyperbolic geometry, linking knot theory with complex hyperbolic orbifolds.

Contribution

It establishes the existence and uniqueness of a spherical CR structure on the figure eight knot complement with unipotent peripheral holonomy.

Findings

The figure eight knot complement admits a uniformizable spherical CR structure.

This structure is unique under the condition of unipotent peripheral holonomy.

The structure relates the knot complement to a complex hyperbolic orbifold.

Abstract

We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.