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Families of knots that cannot be made Legendrian parametrically
Abstract
The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the n>1 case. We prove that for every n≥ 3, every knot type 𝒦, every Legendrian representative ℒ and every formal Legendrian representative ℱℒ, the associated group homomorphisms πn(ℒ)→πn(𝒦) and πn(ℱℒ)→πn(𝒦) are never surjective. We then show that surjectivity at the π2-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond π1.
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BibTeX
@misc{martinezaguinaga2026families,
title = {{Families of knots that cannot be made Legendrian parametrically}},
author = {Javier Martínez-Aguinaga},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18492},
eprint = {2609.18492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18492},
}
RIS
TY - UNPB
TI - Families of knots that cannot be made Legendrian parametrically
AU - Javier Martínez-Aguinaga
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18492
UR - https://arxiv.org/abs/2609.18492
AB - The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the n>1 case. We prove that for every n≥ 3, every knot type 𝒦, every Legendrian representative ℒ and every formal Legendrian representative ℱℒ, the associated group homomorphisms π_n(ℒ)→π_n(𝒦) and π_n(ℱℒ)→π_n(𝒦) are never surjective. We then show that surjectivity at the π_2-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond π_1.
ER -