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We compute the homotopy type of the space of embeddings of convex disks withLegendrian boundary into a tight contact $3$-manifold, whenever the sum of theabsolute value of the rotation number of the boundary with theThurston-Bennequin invariant is $-1$, proving that it is homotopy equivalent tothe space of smooth embeddings. Using the same ideas it is also determined thehomotopy type of the space of embeddings of convex spheres into a tight$3$-fold in terms of the space of smooth spheres. As a consequence we determinethe homotopy type of the space of long Legendrian unknots, satisfying theprevious condition, into a tight $3$-fold and also of the space of longtransverse unknots with self-linking number $-1$, proving that these spaces arehomotopy equivalent to the space of smooth long unknots. We also determine thehomotopy type of the contactomorphism group of every universally tighthandlebody, the standard $\NS^1\times\NS^2$ and every Legendrian fibration overa compact orientable surface with non-empty boundary, partially solving aconjecture due to E. Giroux. Finally, we show that the space of embeddings ofLegendrian $(n,n)$-torus links with maximal Thurston-Bennequin invariant ishomotopy equivalent to $\U(2)\times K(\mathcal{M}_n,1)$, where $\mathcal{M}_n$is the mapping class group of the $2$-sphere with $n$-holes.

The homotopy type of the contactomorphism groups of tight contact $3$-manifolds, part I