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Classification of Engel knots


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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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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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).

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Mathematics Subject Classification

  • Primary 53D10
  • Secondary 53D15
  • 57R17