Knots and links of complex tangents

TL;DR

This paper demonstrates that any knot or link can be realized as the set of complex tangents of a smooth embedding of a 3-sphere into complex space, using singularity theory techniques.

Contribution

It establishes a complete characterization of knots and links that can appear as complex tangents in embeddings of 3-manifolds, introducing new applications of singularity theory.

Findings

Any knot or link can be realized as complex tangents of a 3-sphere embedding.

A 1-dimensional submanifold can be realized as complex tangents iff it is null-homologous.

The proof employs novel applications of singularity theory of differentiable maps.

Abstract

It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised as the set of complex tangents of a smooth embedding of the 3-manifold into the three-dimensional complex space if and only if it represents the trivial integral homology class in the 3-manifold. The proof involves a new application of singularity theory of differentiable maps.