On the Topological Structure Of Complex Tangencies to Embeddings of S³ into ℂ³

2015/06/26 by Ali M. Elgindi, Elgindi, Ali M.
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #math.CV #math.GT

paper · pdf · doi:10.48550/arxiv.1506.07992

Formal version published at: New York J. of Math. Vol. 18 (2012), 295-313

arxiv created 2015/06/26 · openalex publication_date 2015/06/26 · arxiv updated 2015/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the mid-1980's, M. Gromov used his machinery of the h-principle to prove that there exists totally real embeddings of S³ into ℂ³. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, namely where complex tangents arise as codimension-2 subspaces. We first consider the Heisenberg group ℍ and generate some interesting results there-in. Then, by using the biholomorphism of ℍ with the 3-sphere minus a point, we demonstrate that every homeomorphism-type of knot in S³ may arise precisely as the set of complex tangents to an embedding S³ \hookrightarrow ℂ³. We also make note of the (non-generic) situation where complex tangents arise along surfaces.

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