Motion groups, Legendrian links, and non-trivial loops of embeddings

We introduce, for an embedded Legendrian link~$L$ in a contact 3-manifold~$(M,ξ)$, the group of its \emph{Legendrian motions}. Analogously with the smooth case, we prove that this Legendrian motion group is isomorphic to the fundamental group of the space of unparametrised Legendrian embeddings. For