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 $n\geq 3$, every knot type $\mathcal K$, every Legendrian representative $\mathcal L$ and every formal Legendrian representative $\mathcal{FL}$, the associated group homomorphisms $π_n(\mathcal{L})\toπ_n(\mathcal{K})$ and $π_n(\mathcal{FL})\toπ_n(\mathcal{K})$ 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 $π_1$.