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.References
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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
- MR Author ID: 1037602
- 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
- MR Author ID: 645634
- Email: mtakase@st.seikei.ac.jp
- 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.
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3739200