Augmentations and Legendrians at the IAS

Satellite operations are a common way to create interesting knot types in the smooth category. It starts with a knot $K$, called the companion knot, in some manifold $M$ and another knot $P$, called the pattern, in $S^1 \times D^2$ and then creates a third knot $P(K)$, called the satellite knot, as the image of $P$ when $S^1 \times D^2$ is identified with a neighborhood of $K$. In this talk we will discuss the relation between Legendrian knots representing $K$, $P$, and $P(K)$. Sometimes the classification of Legendrian representatives for $K$ and $P$ yields a classification for $P(K)$, but other times it does not. We will discuss why this happens and a general framework for studying Legendrian Satellites.