On the Legendrian realisation of loops of knots

Abstract

We study the natural inclusion of the space of Legendrian embeddings in $({\mathbb{S}}^3,{\xi }_{\mathrm{std}})$ into the space of smooth embeddings at the level of fundamental groups. T. Kálmán posed in [Kal] the open question of whether for every fixed knot type $\mathcal{K}$ and Legendrian representative $\mathcal{L}$, the homomorphism ${\pi }_1(\mathcal{L})\to {\pi }_1(\mathcal{K})$ is surjective. We positively answer this question for infinitely many knot types $\mathcal{K}$ in the three main families (hyperbolic, torus and satellites) and every stabilised Legendrian representative in $({\mathbb{S}}^3,{\xi }_{\mathrm{std}})$.

Revised version. The title of the article has been changed from `On the Legendrian realisability of parametric families of knots'' to "On the Legendrian realisability of loops of knots". Additionally, the results concerning higher homotopy groups in v1 have been split off into a separate project. 14 pages, 5 figures