Metrically Un-knotted Corank 1 Singularities of Surfaces in \(\mathbb {R}^4\)

Abstract

The paper is devoted to relations between topological and metric properties of germs of real surfaces, obtained by analytic maps from \(\mathbb {R}^2\) to \(\mathbb {R}^4.\) We show that for a big class of such surfaces, the normal embedding property implies the triviality of the knot, presenting the link of the surfaces. We also present some criteria of normal embedding in terms of the polar curves.

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

References

  1. 1.

    Birbrair, L., Fernandes, A.C.G.: Metric theory of semialgebraic curves. Rev. Mat. Complut. 13(2), 369–382 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  2. 3.

    Birbrair, L., Mostowski, T.: Normal embeddings of semialgebraic sets. Mich. Math. J. 47, 125–132 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  3. 4.

    Birbrair, L., Mendes, R.: Arc criterion of normal embedding. In: Advances in Singularities and Foliations: Geometry, Topology and Applications. Springer Proceedings in Mathematics & Statistics (2018)

  4. 5.

    Fernandes, A.: Topological equivalence of complex curves and bi-Lipschitz homeomorphisms. Mich. Math. J. 51(3), 593–606 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  5. 6.

    Fox, R.H., Milnor, J.W.: Singularities of 2-spheres in 4-space and cobordism of knots. Osaka J. Math. 3, 257–267 (1966)

    MathSciNet  MATH  Google Scholar 

  6. 7.

    Marar, W.L., Mond, D.: Multiple point schemes for corank 1 maps. J. Lond. Math. Soc. 39, 553–567 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  7. 8.

    Kurdyka, K., Orro, P.: Distance géodésique sur un sous-analytique. Rev. Mat. Univ. Complut. Madr. 10(Suplementario), 173–182 (1997)

  8. 9.

    Marar, W.L., Nuño-Ballesteros, J.J.: The doodle of a finitely determined map germ from \({\mathbb{R}}^2\) to \({\mathbb{R}}^3\). Adv. Math. 221, 1281–1301 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  9. 10.

    Mendes, R., Nuño-Ballesteros, J.J.: Knots and the topology of singular surfaces in \({\mathbb{R}}^4\). In: Real and Complex Singularities. Contemporary Mathematics, vol. 675, pp. 229–239. American Mathematical Society, Providence (2016)

  10. 11.

    Mendes, R.: Geometria métrica e topologia de superfícies algebricamente parametrizadas. Tese de doutorado, UFC (2016)

    Google Scholar 

  11. 12.

    O’Shea, D.B., Wilson, L.C.: Exceptional rays and bilipschitz geometry of real surface singularities. Topol. Appl. 234, 359–374 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  12. 13.

    Teissier, F., Pham, F.: Fractions lipschitzi-ennes d’une algébre analytique complexe et saturation de Zariski. Centre de Mathématiques de l’Ecole Polytechnique (Paris), June 1969

Download references

Acknowledgements

We would like to thank Alexandre Fernandes, Vincent Grandjean, and Edson Sampaio for interesting discussions and important remarks.

Author information

Affiliations

Authors

Corresponding author

Correspondence to L. Birbrair.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Birbrair, L., Mendes, R. & Nuño-Ballesteros, J.J. Metrically Un-knotted Corank 1 Singularities of Surfaces in \(\mathbb {R}^4\). J Geom Anal 28, 3708–3717 (2018). https://doi.org/10.1007/s12220-017-9973-2

Download citation

Keywords

  • Normal embedding
  • Link
  • Isolated singularity

Mathematics Subject Classification

  • 14B05
  • 32S50
  • 58K15