Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Patrick Ahern and Walter Rudin, Totally real embeddings of $ S^3$ in $ {\bf C}^3$, Proc. Amer. Math. Soc. 94 (1985), no. 3, 460-462. MR 787894, https://doi.org/10.2307/2045235
  • [2] S. Akbulut and H. King, All knots are algebraic, Comment. Math. Helv. 56 (1981), no. 3, 339-351. MR 639356, https://doi.org/10.1007/BF02566217
  • [3] Michèle Audin, Fibrés normaux d'immersions en dimension double, points doubles d'immersions lagragiennes et plongements totalement réels, Comment. Math. Helv. 63 (1988), no. 4, 593-623 (French). MR 966952, https://doi.org/10.1007/BF02566781
  • [4] Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-21. MR 0200476
  • [5] Eduardo Chincaro, Bifurcations of Whitney maps, Ph.D. thesis, Tese de doutorado, IMPA (1978).
  • [6] A. V. Domrin, A description of characteristic classes of real submanifolds in complex manifolds in terms of RC-singularities, Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), no. 5, 19-40 (Russian, with Russian summary); English transl., Izv. Math. 59 (1995), no. 5, 899-918. MR 1360632, https://doi.org/10.1070/IM1995v059n05ABEH000039
  • [7] Ali M. Elgindi, On the topological structure of complex tangencies to embeddings of $ S^3$ into $ \mathbb{C}^3$, New York J. Math. 18 (2012), 295-313. MR 2928578
  • [8] Ali M. Elgindi, A topological obstruction to the removal of a degenerate complex tangent and some related homotopy and homology groups, Internat. J. Math. 26 (2015), no. 5, 1550025, 16. MR 3345506
  • [9] Ali M. Elgindi, Totally real perturbations and nondegenerate embeddings of $ S^3$, New York J. Math. 21 (2015), 1283-1293. MR 3441643
  • [10] Y. Eliashberg and N. Mishachev, Introduction to the $ h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245
  • [11] C. Ernst and D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 3, 489-515. MR 1068451, https://doi.org/10.1017/S0305004100069383
  • [12] Franc Forstnerič, On totally real embeddings into $ {\bf C}^n$, Exposition. Math. 4 (1986), no. 3, 243-255. MR 880125
  • [13] Franc Forstnerič, Complex tangents of real surfaces in complex surfaces, Duke Math. J. 67 (1992), no. 2, 353-376. MR 1177310, https://doi.org/10.1215/S0012-7094-92-06713-5
  • [14] Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505
  • [15] André Haefliger, Quelques remarques sur les applications différentiables d'une surface dans le plan, Ann. Inst. Fourier. Grenoble 10 (1960), 47-60 (French). MR 0116357
  • [16] León Kushner, Harold Levine, and Paulo Porto, Mapping three-manifolds into the plane. I, Bol. Soc. Mat. Mexicana (2) 29 (1984), no. 1, 11-33. MR 790729
  • [17] Hon Fei Lai, Characteristic classes of real manifolds immersed in complex manifolds, Trans. Amer. Math. Soc. 172 (1972), 1-33. MR 0314066, https://doi.org/10.2307/1996329
  • [18] Harold Levine, Classifying immersions into $ {\bf R}^4$ over stable maps of $ 3$-manifolds into $ {\bf R}^2$, Lecture Notes in Mathematics, vol. 1157, Springer-Verlag, Berlin, 1985. MR 814689
  • [19] Harold Levine, Stable mappings of $ 3$-manifolds into the plane, Singularities (Warsaw, 1985) Banach Center Publ., vol. 20, PWN, Warsaw, 1988, pp. 279-289. MR 1101845
  • [20] J. N. Mather, Stability of $ C^{\infty}$ mappings. VI: The nice dimensions, Proceedings of Liverpool Singularities-Symposium, I (1969/70), Lecture Notes in Math., Vol. 192, Springer, Berlin, 1971, pp. 207-253. MR 0293670
  • [21] John N. Mather, Generic projections, Ann. of Math. (2) 98 (1973), 226-245. MR 0362393, https://doi.org/10.2307/1970783
  • [22] Colin Rourke and Brian Sanderson, The compression theorem. I, Geom. Topol. 5 (2001), 399-429. MR 1833749, https://doi.org/10.2140/gt.2001.5.399
  • [23] Osamu Saeki, Constructing generic smooth maps of a manifold into a surface with prescribed singular loci, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 4, 1135-1162 (English, with English and French summaries). MR 1359844
  • [24] Osamu Saeki, Simple stable maps of $ 3$-manifolds into surfaces, Topology 35 (1996), no. 3, 671-698. MR 1396772, https://doi.org/10.1016/0040-9383(95)00034-8
  • [25] Osamu Saeki and Masamichi Takase, Desingularizing special generic maps, J. Gökova Geom. Topol. GGT 7 (2013), 1-24. MR 3153918
  • [26] J. Sotomayor, Bifurcation of whitney maps $ \mathbf {R}^n\to \mathbf {R}^2$ and critical pareto sets, Proceedings of a Symposium titled ``Applications of Topology and Dynamical Systems'' held at the University of Warwick, Coventry, 1973/1974.
  • [27] Masamichi Takase, An Ekholm-Szűcs-type formula for codimension one immersions of 3-manifolds up to bordism, Bull. Lond. Math. Soc. 39 (2007), no. 1, 39-45. MR 2303517, https://doi.org/10.1112/blms/bdl009
  • [28] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble 6 (1955-1956), 43-87 (French). MR 0087149
  • [29] S. M. Webster, The Euler and Pontrjagin numbers of an $ n$-manifold in $ {\bf C}^n$, Comment. Math. Helv. 60 (1985), no. 2, 193-216. MR 800003, https://doi.org/10.1007/BF02567410
  • [30] Wu Wen-tsün, On the immersion of $ C^{\infty}$-$ 3$-manifolds in a Euclidean space, Sci. Sinica 13 (1964), 335-336. MR 0172302
  • [31] Minoru Yamamoto, First order semi-local invariants of stable maps of 3-manifolds into the plane, Proc. London Math. Soc. (3) 92 (2006), no. 2, 471-504. MR 2205725, https://doi.org/10.1112/S0024611505015534

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32V40, 57M25, 57R45, 57R40, 53C40

Retrieve articles in all journals with MSC (2010): 32V40, 57M25, 57R45, 57R40, 53C40


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

American Mathematical Society