On the Legendrian realisation of parametric families of knots

We study the natural inclusion of the space of Legendrian embeddings in (S³, ₒₓ₃) 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 ₁ (L) ₁ (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, ₒₓ₃). We then show that for every n 3, the homomorphisms ₙ (L) ₙ (K) and ₙ (FL) ₙ (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 ₃. For completeness, we also show that surjectivity at the ₂-level depends on the smooth knot type.