Families of knots that cannot be made Legendrian parametrically

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 $n\ge 3$, every knot type $\mathcal{K}$, every Legendrian representative $\mathcal{L}$ and every formal Legendrian representative $\mathcal{F}\mathcal{L}$, the associated group 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. We then show that surjectivity at the ${\pi }_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond ${\pi }_1$.