We determine the homotopy type of the spaces of several Legendrian knots and links with the maximal Thurston–Bennequin invariant. In particular, we give a recursive formula of the homotopy type of the space of Legendrian embeddings of sufficiently positive cables, and determine the homotopy type of the space of Legendrian embeddings of Seifert fibered links, which include all torus knots and links, in the standard contact 3-sphere, except when one of the link components is a negative torus knot. In general, we prove that the space of contact structures on the complement of a sufficiently positive Legendrian cable with the maximal Thurston–Bennequin invariant is homotopy equivalent to the space of contact structures on the complement of the underlying Legendrian knot, and prove that the space of contact structures on a Legendrian Seifert fibered space over a compact oriented surface with boundary is contractible. From this result, we find infinitely many new components of the space of Legendrian embeddings in the standard contact 3-sphere that satisfy an injective h -principle. These include the spaces of Legendrian embeddings of an algebraic link with the maximal Thurston–Bennequin invariant. In particular, the inclusion of these Legendrian embedding spaces into the corresponding formal Legendrian embedding spaces is a homotopy injection.
Fernández et al. (Sun,) studied this question.