Surjectivity of the natural inclusion on fundamental groups for arbitrary knots

Determine whether the homomorphism on fundamental groups induced by the natural inclusion from the space of Legendrian embeddings in the standard contact three-sphere to the space of smooth embeddings of the circle in the three-sphere is surjective for every knot type.

Background

The paper considers the relationship between the homotopy groups of the space of Legendrian embeddings in the standard contact three-sphere and those of the corresponding space of smooth knot embeddings. At the level of path components, every smooth knot type has a Legendrian representative, but analogous surjectivity questions for higher homotopy groups are more restrictive.

The unresolved problem concerns the case of the fundamental group. The natural inclusion from Legendrian embeddings to smooth embeddings induces a homomorphism on \pi_1, and although surjectivity has been established for infinitely many knots in the hyperbolic, torus, and satellite families, the general case remains unresolved.

References

T. Kálmán posed this precise question (see p. 2016) at the level of $\pi_1$; i.e. he asked whether the $\pi_1$-homomorphism induced by the natural inclusion \begin{equation}\label{NaturalInclusion} i:(3,\xi_) \hookrightarrow (1,3). \end{equation}

was always surjective. A positive answer to this question has been provided for infinitely many knots in the three main families (hyperbolic, torus and satellites) in the recent preprint but the question remains open in the general case since first proposed in 2005.

NaturalInclusion:

i:(3,ξ)↪(1,3).i:(^3,\xi_) \hookrightarrow (^1,^3).

— Families of knots that cannot be made Legendrian parametrically  (2609.18492 - Martínez-Aguinaga, 16 Sep 2026) in Section 1, Introduction