In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of S³ into C³. In particular, given any embedding of S³ and a neighborhood of the complex tangents of the embedding, we show that there exists a (C⁰-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 ⊂ S³, either there exists an embedding of S³ which assumes nondegenerate complex tangents exactly along K or there exists a nondegenerate embedding complex tangent along two unlinked copies of K (both cases may hold). We also note possible directions of future investigations.

The author was supported in part by IRG grant no. 4010501011418 from Alfaisal University.