Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3708–3717 | Cite as

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

  • L. BirbrairEmail author
  • Rodrigo Mendes
  • J. J. Nuño-Ballesteros
Article

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.

Keywords

Normal embedding Link Isolated singularity 

Mathematics Subject Classification

14B05 32S50 58K15 

Notes

Acknowledgements

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

References

  1. 1.
    Birbrair, L., Fernandes, A.C.G.: Metric theory of semialgebraic curves. Rev. Mat. Complut. 13(2), 369–382 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 3.
    Birbrair, L., Mostowski, T.: Normal embeddings of semialgebraic sets. Mich. Math. J. 47, 125–132 (2000)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  4. 5.
    Fernandes, A.: Topological equivalence of complex curves and bi-Lipschitz homeomorphisms. Mich. Math. J. 51(3), 593–606 (2003)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  6. 7.
    Marar, W.L., Mond, D.: Multiple point schemes for corank 1 maps. J. Lond. Math. Soc. 39, 553–567 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 8.
    Kurdyka, K., Orro, P.: Distance géodésique sur un sous-analytique. Rev. Mat. Univ. Complut. Madr. 10(Suplementario), 173–182 (1997)Google Scholar
  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)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  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)MathSciNetCrossRefzbMATHGoogle 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 1969Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  • L. Birbrair
    • 1
    Email author
  • Rodrigo Mendes
    • 2
  • J. J. Nuño-Ballesteros
    • 3
  1. 1.Departamento of MatemáticaUniversidade Federal do CearáFortalezaBrazil
  2. 2.Instituto de Ciências Exatas e da NaturezaUniversidade de Integração Internacional da Lusofonia Afro-Brasileira (Unilab)AcarapeBrazil
  3. 3.Departament de MatemàtiquesUniversitat de ValènciaBurjassotSpain

Personalised recommendations