Classification of Engel knots

Abstract

Let \((M,{\mathcal {D}})\) be an Engel 4-manifold. We show that the scanning map from the space of Engel knots to the space of formal Engel knots is a weak homotopy equivalence when restricted to the complement of the closed \(\ker ({\mathcal {D}})\)-orbits. This is a relative, parametric, and \(C^0\)-close h-principle.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Adachi, J.: Classification of horizontal loops in standard Engel space. Int. Math. Res. Not. 2007, 29 (2007). https://doi.org/10.1093/imrn/rnm008

  2. 2.

    Arnol’d, V.I., Givental, A.: Symplectic Geometry, Dynamical Systems IV, Encyc. M. S. Springer, Berlin (2001)

    Google Scholar 

  3. 3.

    Borman, M.S., Eliashberg, Y., Murphy, E.: Existence and classification of overtwisted contact structures in all dimensions. Acta Math. 215(2), 281–361 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bryant, R.L., Hsu, L.: Rigidity of integral curves of rank 2 distributions. Invent. Math. 114(2), 435–461 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Cartan, E.: Sur quelques quadratures dont l’élément différentiel contient des fonctions arbitraires. Bull. Soc. Math. Fr. 29, 118–130 (1901)

    Article  MATH  Google Scholar 

  6. 6.

    Casals, R., Pérez, J.L., del Pino, Á., Presas, F.: Existence h-principle for Engel structures. Invent. Math. 210(2), 417–451 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Casals, R., del Pino, Á., Presas, F.: Loose Engel structures. arXiv:1712.09283

  8. 8.

    Chekanov, Y.: Differential algebra of Legendrian links. Invent. Math. 150(3), 441–483 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    del Pino, Á., Presas, F.: Flexibility for tangent and transverse immersions in Engel manifolds. arXiv:1609.09306

  10. 10.

    del Pino, Á., Vogel, T.: Overtwisted Engel structures (in preparation)

  11. 11.

    Eliashberg, Y., Mishachev, N.: Introduction to the h-principle. Graduate Studies in Mathematics, vol. 48. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  12. 12.

    Eliashberg, Y., Mishachev, N.: Wrinkled Embeddings. Foliations, Geometry, and Topology, pp. 207–232, Contemp. Math., vol. 498. American Mathematical Society, Providence (2009)

  13. 13.

    Eliashberg, Y., Kotschick, D., Murphy, E., Vogel, T.: Engel Structures: Workshop Report. American Institute of Mathematics, San Jose (2017)

    Google Scholar 

  14. 14.

    Fernández, E., Martínez-Aguinaga, J., Presas, F.: Fundamental groups of formal Legendrian and horizontal embedding spaces. arXiv:1711.04320

  15. 15.

    Frobenius, G.: Über das Pfaffsche Problem. J. für Reine und Agnew. Math. 82, 230–315 (1877)

    MathSciNet  Google Scholar 

  16. 16.

    Fuchs, D., Tabachnikov, S.: Invariants of Legendrian and transverse knots in the standard contact space. Topology 5, 1025–1053 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Geiges, H.: An Introduction to Contact Topology. Cambridge Stud. Adv. Math., vol. 109. Cambridge University Press, Cambridge (2008)

  18. 18.

    Geiges, H.: Horizontal loops in Engel space. Math. Ann. 342(2), 291–296 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Gromov, M.: Partial Differential Relations. Ergebnisse Math. und ihrer Gren. Springer, Berlin (1986)

    Book  Google Scholar 

  20. 20.

    Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics. Springer, Berlin (1974)

    Google Scholar 

  21. 21.

    Montgomery, R.: Engel deformations and contact structures. Northern California Symplectic Geometry Seminar, pp. 103–117. Amer. Math. Soc. Transl. Ser. 2, vol. 196. American Mathematical Society, Providence (1999)

  22. 22.

    McDuff, D.: Applications of convex integration to symplectic and contact geometry. Ann. Inst. Fourier 37, 107–133 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Murphy, E.: Loose Legendrian embeddings in high dimensional contact manifolds. arXiv:1201.2245

  24. 24.

    Segal, G.: The topology of spaces of rational functions. Acta Math. 143(1—-2), 3972 (1979)

    MathSciNet  Google Scholar 

  25. 25.

    Vogel, T.: Existence of Engel structures. Ann. Math. (2) 169(1), 79–137 (2009)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to the American Institute of Mathematics for sponsoring the workshop Engel Structures on April 2017, where this was one of the questions of interest; we are pleased to acknowledge E. Murphy, F. Presas, L. Traynor and T. Vogel for useful discussions. R. Casals is supported by the NSF Grant DMS-1608018 and a BBVA Research Fellowship, and Á. del Pino is supported by the Grant NWO Vici Grant No. 639.033.312 of M. Crainic.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Roger Casals.

Additional information

Communicated by Jean-Yves Welschinger.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Casals, R., del Pino, Á. Classification of Engel knots. Math. Ann. 371, 391–404 (2018). https://doi.org/10.1007/s00208-017-1625-0

Download citation

Mathematics Subject Classification

  • Primary 53D10
  • Secondary 53D15
  • 57R17