Fundamental groups of formal Legendrian and horizontal embedding spaces

Fernández, Eduardo; Martínez-Aguinaga, Javier; Presas, Francisco

doi:10.2140/agt.2020.20.3219

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Fundamental groups of formal Legendrian and horizontal embedding spaces

Eduardo Fernández, Javier Martínez-Aguinaga and Francisco Presas

Algebraic & Geometric Topology 20 (2020) 3219–3312

DOI: 10.2140/agt.2020.20.3219

Abstract

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in $ℝ^{3}$ . We use it to show that previous examples in the literature of nontrivial loops of Legendrian embeddings are already nontrivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel $ℝ^{4}$ . We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at $π_{1}$ –level.

Keywords

Legendrian knots, horizontal knots, formal embeddings

Mathematical Subject Classification

Primary: 57R17, 58A17

Secondary: 53D10

References

Publication

Received: 27 November 2017

Revised: 11 November 2019

Accepted: 9 December 2019

Published: 29 December 2020

Authors

Eduardo Fernández
Departamento de Álgebra, Geometría y Topología, Facultad de Matemáticas, Universidad Complutense de Madrid, and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Madrid Spain

Javier Martínez-Aguinaga
Departamento de Álgebra, Geometría y Topología, Facultad de Matemáticas, Universidad Complutense de Madrid, and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Madrid Spain

Francisco Presas
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Madrid Spain

Volume 20, issue 7 (2020)