#### Volume 20, issue 7 (2020)

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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
##### 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 ${\pi }_{1}$–level.

##### Keywords
Legendrian knots, horizontal knots, formal embeddings
##### Mathematical Subject Classification
Primary: 57R17, 58A17
Secondary: 53D10
##### Publication
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