Skip to main content
SpringerLink
Log in

On real algebraic links in the 3-sphere associated with mixed polynomials

  • Research
  • Published:
Research in the Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the 3-sphere. Especially, we describe this construction in the case of semiholomorphic polynomials, which are mixed polynomials that are holomorphic in one variable. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti–Shiota conjecture, which states that any fibered link on the 3-sphere is a real algebraic link.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Availability of data and materials

Not applicable.

Notes

  1. Consequently, it is analytically equivalent to a holomorphic polynomial function.

  2. \(B=w^2\) for a braid w on s strands.

References

  1. Akbulut, S., King, H.: All knots are algebraic. Comment. Math. Helv. 56(3), 339–351 (1981). https://doi.org/10.1007/BF02566217

  2. Araújo dos Santos, R.N., Bode, B., Sanchez Quiceno, E.L.: Links of singularities of inner non-degenerate mixed functions. arXiv:2208.11655 (2022)

  3. Benedetti, R., Shiota, M.: On real algebraic links in \(S^3\). Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1(3), 585–609 (1998)

    MathSciNet  Google Scholar 

  4. Blanlœil, V., Oka, M.: Topology of strongly polar weighted homogeneous links. SUT J. Math. 51(1), 119–128 (2015)

    Article  MathSciNet  Google Scholar 

  5. Bode, B., Sanchez Quiceno E.L.: Inner and Partial non-degeneracy of mixed functions. arXiv:2307.15340 (2023)

  6. Bode, B.: Closures of T-homogeneous braids are real algebraic. arXiv:2211.15394 (2022)

  7. Bode, B.: Knotted fields and real algebraic links. Ph.D. thesis, School of Physics, University of Bristol (2018)

  8. Bode, B.: Links of singularities of inner non-degenerate mixed functions, Part II. arXiv:2307.15340 (2022)

  9. Bode, B.: Twisting and satellite operations on p-fibered braids. Commun. Anal. Geom. (2021)

  10. Bode, B.: Constructing links of isolated singularities of polynomials \({\mathbb{R} }^4\rightarrow {\mathbb{R} }^2\). J. Knot Theory Ramifications 28(1), 1950009–21 (2019). https://doi.org/10.1142/S0218216519500093

    Article  MathSciNet  Google Scholar 

  11. Bode, B.: Real algebraic links in \(S^3\) and braid group actions on the set of n-adic integers. J. Knot Theory Ramifications 29(6), 2050039–44 (2020). https://doi.org/10.1142/S021821652050039X

    Article  MathSciNet  Google Scholar 

  12. Bode, B.: All links are semiholomorphic. Eur. J. Math. 9(3), 85 (2023). https://doi.org/10.1007/s40879-023-00678-1

    Article  MathSciNet  Google Scholar 

  13. Bode, B., Dennis, M.R.: Constructing a polynomial whose nodal set is any pre- scribed knot or link. J. Knot Theory Ramifications 28(1), 1850082–31 (2019). https://doi.org/10.1142/S0218216518500827

    Article  MathSciNet  Google Scholar 

  14. Brieskorn, E., Knörrer, H.: Local investigations. Birkhäuser Basel, Basel, , pp. 325–575 (1986). https://doi.org/10.1007/978-3-0348-5097-18

  15. Dennis, M.R., Bode, B.: Constructing a polynomial whose nodal set is the three- twist knot 52. J. Phys. A 50(26), 265204–18 (2017). https://doi.org/10.1088/1751-8121/aa6cbe

    Article  ADS  MathSciNet  Google Scholar 

  16. Inaba, K., Kawashima, M., Oka, M.: Topology of mixed hypersurfaces of cyclic type. J. Math. Soc. Jpn. 70(1), 387–402 (2018). https://doi.org/10.2969/jmsj/07017538

    Article  MathSciNet  Google Scholar 

  17. Looijenga, E.: A note on polynomial isolated singularities. Nederl. Akad. Weten- sch. Proc. Ser. A 74=Indag. Math. 33, 418–421 (1971)

  18. Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, No. 61, p. 122. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1968)

  19. Oka, M.: Mixed functions of strongly polar weighted homogeneous face type. In: Singularities in Geometry and Topology 2011. Adv. Stud. Pure Math., vol. 66, pp. 173–202. Math. Soc. Japan, Tokyo (2015). https://doi.org/10.2969/aspm/06610173

  20. Oka, M.: Topology of polar weighted homogeneous hypersurfaces. Kodai Math. J. 31(2), 163–182 (2008). https://doi.org/10.2996/kmj/1214442793

    Article  MathSciNet  Google Scholar 

  21. Oka, M.: Non-degenerate mixed functions. Kodai Math. J. 33(1), 1–62 (2010). https://doi.org/10.2996/kmj/1270559157

    Article  MathSciNet  Google Scholar 

  22. Perron, B.: Le n œud “huit” est algébrique réel. Invent. Math. 65(3), 441–451 (1981/82). https://doi.org/10.1007/BF01396628

  23. Pichon, A.: Real analytic germs \(f{\bar{g}}\) and open-book decompositions of the 3-sphere. Int. J. Math. 16(1), 1–12 (2005). https://doi.org/10.1142/S0129167X05002710

    Article  Google Scholar 

  24. Rudolph, L.: Isolated critical points of mappings from \({\mathbb{R}}^4 \rightarrow {\mathbb{R}}^2\) and a natural splitting of the Milnor number of a classical fibered link. I. Basic theory; examples. Comment. Math. Helv. 62(4), 630–645 (1987). https://doi.org/10.1007/BF0256446722

    Article  MathSciNet  Google Scholar 

  25. Zariski, O.: On the topology of algebroid singularities. Am. J. Math. 54(3), 453–465 (1932). https://doi.org/10.2307/2370887

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are immensely grateful to Professor Osamu Saeki for discussions and encouragements and to Benjamin Bode for the comments and recommendations. Additionally, we would like to express our appreciation for the valuable comments and suggestions provided by the referees. Their insights greatly contributed to the improvement of our work. Part of this paper was developed during the visit of second author at the Kyushu University, Japan.

Funding

This work has been supported by São Paulo Research Foundation (FAPESP) agency, grants: 2019/21181-0 thematic research project “New frontiers in Singularity Theory”, the fellowships BEPE/Fapesp 2019/11415-3 and the regular Ph.D fellowship 2017/25902-8.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science (ICMC), University of São Paulo (USP), Avenida Trabalhador São-carlense, 400 - Centro, São Carlos, São Paulo, 13566-590, Brazil

    Raimundo N. Araújo dos Santos & Eder L. Sanchez Quiceno

  2. Departamento de Matemática, Universidade Federal do Ceará, Av. Humberto Monte, s/n Campus do Pici - Bloco 914, Fortaleza, Ceará, 60455-760, Brazil

    Eder L. Sanchez Quiceno

Authors
  1. Raimundo N. Araújo dos Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eder L. Sanchez Quiceno
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eder L. Sanchez Quiceno.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Araújo dos Santos, R.N., Sanchez Quiceno, E.L. On real algebraic links in the 3-sphere associated with mixed polynomials. Res Math Sci 11, 22 (2024). https://doi.org/10.1007/s40687-024-00424-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-024-00424-3

Keywords

Mathematics Subject Classification

Search

Navigation