Satellite operation and Legendrian knots

Legendrian knots are knots whose height above the (x,y)-plane is given by the tangent of the projection dx/dy. These knots naturally arise as trajectories of objects in phase spaces, and they also give one of the fundamental tools to study contact structures. Legendrian classification refines the usual classification of smooth knots; and this refinement have been completely understood for some smooth knot types including the unknot, figure eight knot, torus knots, twist knots and connected sums of these. In this talk, after explaining the background and history of classification of Legendrian knots I will explain a structural result for the satellite operation and give Legendrian classification in some satellite knot types. This is a joint work with John Etnyre.