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Classification of Engel knots

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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.

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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.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Roger Casals

  2. Department of Mathematics, Utrecht University, Budapestlaan 6, 3584, Utrecht, The Netherlands

    Álvaro del Pino

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  1. Roger Casals
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  2. Álvaro del Pino
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Correspondence to Roger Casals.

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Communicated by Jean-Yves Welschinger.

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

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