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

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.