A spherical CR structure on the complement of the figure eight knot with discrete holonomy

By Elisha Falbel. 2005

64 citations recorded by the source corpus.

We describe a general geometrical construction of spherical CR structures. We construct then spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in $PU(2,1,{\bf Z}[\omega])$ and $PU(2,1,{\bf Z}[i])$ respectively. These are the same ring of integers appearing in the real hyperbolic geometry of the corresponding links.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b6c08e3639a779cdeb064957123f9d2e98a2f840

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In Applied Mathematics

1. Martin Deraux
2. C. Zickert
3. A. Antonin
4. Matthias Goerner
5. Nicolas Bergeron
6. Stavros Garoufalidis
7. William M. Goldman
8. Elisha Falbel

In Geometry and Topology

1. Stavros Garoufalidis
2. Matthias Goerner
3. C. Zickert
4. Martin Deraux
5. A. Antonin
6. Nicolas Bergeron
7. William M. Goldman
8. Elisha Falbel

Citations in the corpus

• Tetrahedra of flags, volume and homology of SL.3/ (2011, methodology): “The spherical CR structures with unipotent boundary holonomy were obtained in [5] as the following solutions (up to conjugation): The first solution above corresponds to a discrete representation of the fundamental group of the complement of the figure eight knot in PU(2 , 1) with faithful boundary…”
• Gluing equations for PGL(n, ℂ)–representations of 3–manifolds (2012, methodology): “Shape coordinates for nD 3 have been studied independently by Bergeron, Falbel and Guilloux [1], and were also used by Falbel [8] and Falbel and Wang [11] in connection with spherical CR–structures.”
• Flexing closed hyperbolic manifolds. (2007, background): “(Examples of discrete representations of certain hyperbolic 3–manifold groups into PU.2; 1/ are already known, see Falbel [9] and Schwartz [22].)”
• COMPLEX HYPERBOLIC GEOMETRY OF THE FIGURE EIGHT KNOT (2013, methodology): “Indeed, ordered generic quadruples of points are parametrized up to isometry by appropriate cross ratios, and one can easily write down the corresponding compatibility conditions explicitly [5].”
• Locally Homogeneous Geometric Manifolds (2010, background): “Other interesting examples of spherical CR-structures on 3- manifolds have been constructed by Schwartz [144, 145, 146], Falbel [57], Gusevskii, Parker [137], Parker-Platis [138].”