On the Topological Structure Of Complex Tangencies to Embeddings of $S^3$ into $\mathbb{C}^3$

TL;DR

This paper explores the topological structure of complex tangencies in embeddings of the 3-sphere into complex 3-space, showing that any knot type can appear as the set of complex tangents, with implications for understanding generic and non-generic cases.

Contribution

It demonstrates that every knot type can be realized as the set of complex tangents in embeddings of S^3 into C^3, linking knot theory with complex geometry.

Findings

Any knot in S^3 can be realized as complex tangents of an embedding into C^3.

The Heisenberg group is used to generate examples of complex tangencies.

Non-generic cases include complex tangents along surfaces.

Abstract

In the mid-1980's, M. Gromov used his machinery of the $h$-principle to prove that there exists totally real embeddings of $S^3$ into $\mathbb{C}^3$. 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 $\mathbb{H}$ and generate some interesting results there-in. Then, by using the biholomorphism of $\mathbb{H}$ with the 3-sphere minus a point, we demonstrate that every homeomorphism-type of knot in $S^3$ may arise precisely as the set of complex tangents to an embedding $S^3 \hookrightarrow \mathbb{C}^3$. We also make note of the (non-generic) situation where complex tangents arise along surfaces.