On the Legendrian realisation of parametric families of knots

TL;DR

This paper investigates the homotopical properties of Legendrian knot spaces in the standard contact 3-sphere, proving surjectivity of certain homomorphisms for many knot types and demonstrating rigidity at higher homotopy levels.

Contribution

It provides the first positive results on the surjectivity of homotopy group homomorphisms for Legendrian knots across main knot families and reveals rigidity phenomena at higher homotopy levels.

Findings

Surjectivity of π₁ homomorphisms for many knot types and Legendrian representatives.

Non-surjectivity of πₙ homomorphisms for n ≥ 3 across all knot types.

Dependence of π₂ surjectivity on the smooth knot type.

Abstract

We study the natural inclusion of the space of Legendrian embeddings in $(\mathbb{S}^3,\xi_{\operatorname{std}})$ 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 $\mathcal{K}$ 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 $(\mathbb{S}^3,\xi_{\operatorname{std}})$. We then show that for every $n\geq 3$, the homomorphisms $\pi_n(\mathcal{L})\to\pi_n(\mathcal{K})$ and $\pi_n(\mathcal{FL})\to\pi_n(\mathcal{K})$ are never surjective for any knot type $\mathcal K$, Legendrian representative $\mathcal L$ or formal Legendrian representative $\mathcal{FL}$. 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.