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

TL;DR

This paper constructs a branched spherical CR structure on the figure eight knot complement using a specific holonomy representation, highlighting fundamental differences between two boundary unipotent representations into PU(2,1).

Contribution

It explicitly constructs a branched spherical CR structure for the figure eight knot complement with a particular holonomy, and compares two boundary unipotent representations.

Findings

Explicit construction of the CR structure with holonomy rho_2

Identification of key differences between rho_1 and rho_2 representations

Analysis of the fundamental group's behavior under these representations

Abstract

We obtain a branched spherical CR structure on the complement of the figure eight knot with a given holonomy representation (called rho_2). There are essentially two boundary unipotent representations from the complement of the figure eight knot into PU(2,1), we call them rho_1 and rho_2. 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 rho_1 and rho_2.