Parametric satellites and connected-sums in the space of Legendrian embeddings

This article introduces two new constructions at the higher homotopy level in the space of Legendrian embeddings in (R³, ξstd). We first introduce the parametric Legendrian satellite construction, showing that the satellite operation works for parametric families of Legendrian embeddings. This yields new invariants at the higher-order homotopy level. We then introduce the parametric connected-sum construction. This operation takes as inputs two n-spheres based at Legendrian embeddings K₁ and K₂, respectively, and produces a new n-sphere based at K₁# K₂. As a main application we construct new infinite families of loops of Legendrian embeddings with non-trivial LCH monodromy invariant.