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We determine the homotopy type of the spaces of several Legendrian knots andlinks with the maximal Thurston--Bennequin invariant. In particular, we give arecursive formula of the homotopy type of the space of Legendrian embeddings ofsufficiently positive cables, and determine the homotopy type of the space ofLegendrian embeddings of Seifert fibered links, which include all torus knotsand links, in the standard contact 3-sphere, except when one of the linkcomponents is a negative torus knot. In general, we prove that the space ofcontact structures on the complement of a sufficiently positive Legendriancable with the maximal Thurston-Bennequin invariant is homotopy equivalent tothe space of contact structures on the complement of the underlying Legendrianknot, and prove that the space of contact structures on a Legendrian Seifertfibered space over a compact oriented surface with boundary is contractible.From this result, we find infinitely many new components of the space ofLegendrian embeddings in the standard contact 3-sphere that satisfy aninjective h-principle. These include the spaces of Legendrian embeddings of analgebraic link with the maximal Thurston--Bennequin invariant. In particular,the inclusion of these Legendrian embedding spaces into the correspondingformal Legendrian embedding spaces is a homotopy injection.

Spaces of Legendrian cables and Seifert fibered links