On the Legendrian realisation of parametric families of knots

We study the natural inclusion of the space of Legendrian embeddings in $({\mathbb{S}}^3,{\xi }_{\mathrm{std}})$ into the space of smooth embeddings from a homotopical viewpoint. 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 $({\mathbb{S}}^3,{\xi }_{\mathrm{std}})$. We then show that for every $n\ge 3$, the homomorphisms ${\pi }_n(\mathcal{L})\to {\pi }_n(\mathcal{K})$ and ${\pi }_n(\mathcal{F}\mathcal{L})\to {\pi }_n(\mathcal{K})$ are never surjective for any knot type $\mathcal{K}$, Legendrian representative $\mathcal{L}$ or formal Legendrian representative $\mathcal{F}\mathcal{L}$. This shows the existence of rigidity at every higher-homotopy level beyond ${\pi }_3$. For completeness, we also show that surjectivity at the ${\pi }_2$-level depends on the smooth knot type.