Knots and links of complex tangents

Kasuya, Naohiko; Takase, Masamichi

doi:10.1090/tran/7164

Knots and links of complex tangents
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by Naohiko Kasuya and Masamichi Takase PDF

Trans. Amer. Math. Soc. 370 (2018), 2023-2038 Request permission

Abstract:

It is shown that every knot or link is the set of complex tangents of a $3$-sphere smoothly embedded in the $3$-dimensional complex space. We show in fact that a $1$-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 $3$-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.

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