On the Legendrian realisation of parametric families of knots

We study the natural inclusion of the space of Legendrian embeddings in (S³,ξstd) into the space of smooth embeddings from a homotopical viewpoint. T. K\'alm\'an posed in [Kal] the open question of whether for every fixed knot type K and Legendrian representative L, the homomorphism π₁(L)→π₁(K) is surjective. We positively answer this question for infinitely many knot types K in the three main families (hyperbolic, torus and satellites) and every stabilised Legendrian representative in (S³,ξstd). We then show that for every n≥ 3, the homomorphisms πₙ(L)→πₙ(K) and πₙ(FL)→πₙ(K) are never surjective for any knot type K, Legendrian representative L or formal Legendrian representative FL. This shows the existence of rigidity at every higher-homotopy level beyond π₃. For completeness, we also show that surjectivity at the π₂-level depends on the smooth knot type.