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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Knots and links of complex tangents


Authors: Naohiko Kasuya and Masamichi Takase
Journal: Trans. Amer. Math. Soc. 370 (2018), 2023-2038
MSC (2010): Primary 32V40, 57M25; Secondary 57R45, 57R40, 53C40
DOI: https://doi.org/10.1090/tran/7164
Published electronically: November 16, 2017
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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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Additional Information

Naohiko Kasuya
Affiliation: School of Social Informatics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258, Japan
Address at time of publication: Department of Mathematics, Kyoto Sangyo University, Kamigamo-Motoyama, Kita-ku, Kyoto 603-8555, Japan
Email: nkasuya@cc.kyoto-su.ac.jp

Masamichi Takase
Affiliation: Faculty of Science and Technology, Seikei University, 3-3-1 Kichijoji-kitamachi, Musashino, Tokyo 180-8633, Japan
Email: mtakase@st.seikei.ac.jp

DOI: https://doi.org/10.1090/tran/7164
Keywords: Complex tangent, totally real, embedding, singularity, stable map, knot, link, Thom polynomial, 3-manifold, band surgery, nullification
Received by editor(s): July 21, 2016
Published electronically: November 16, 2017
Additional Notes: The second-named author was supported in part by the Grant-in-Aid for Scientific Research (C), JP15K04880, Japan Society for the Promotion of Science.
Article copyright: © Copyright 2017 American Mathematical Society

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