Totally Real Perturbations and Non-Degenerate Embeddings of $S^3$

TL;DR

This paper develops methods for locally removing complex tangents in embeddings of $S^3$ into $C^3$, enabling totally real embeddings and exploring embeddings with non-degenerate complex tangents along knots.

Contribution

It introduces techniques for modifying embeddings to eliminate complex tangents and characterizes embeddings with non-degenerate complex tangents along specified knots.

Findings

Existence of $C^0$-close totally real embeddings near any given embedding.

Construction of embeddings with non-degenerate complex tangents along specific knot types.

Possibility of embeddings with complex tangents along unlinked copies of knots.

Abstract

In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of $S^3$ into $\mathbb{C}^3$. In particular, given any embedding of $S^3$ and a neighborhood of the complex tangents of the embedding, we show that there exists a ($C^0$-close) totally real embedding which agrees with the original embedding outside the given neighborhood of the complex tangents. We also demonstrate that given any knot type K in $S^3$, either there exists an embedding of $S^3$ which assumes non-degenerate complex tangents exactly along K or there exists a non-degenerate embedding complex tangent along two unlinked copies of K (both cases may hold). We also note possible directions of future investigations.