# Cecilia Karlsson: Legendrian contact homology II

In this second talk, I will define Chekanov's version of Legendrian contact homology (LCH) for Legendrian knots in R3. I will begin with an example, showing that LCH is more sensitive than the classical invariants. This will use a linearized version of the homology. In the second part of the talk I will focus on the proof that the differential indeed squares to zero, and also say something about invariance under Legendrian Reidemeister moves. This is intended to be a smooth introduction to the next talk, where we will consider Legendrian contact homology defined for Legendrians in arbitrary 1-jet spaces. This case is more delicate, and we have to understand the concept of Gromov compactness for pseudo-holomorphic curves to prove that we get a differential graded algebra associated to each Legendrian, whose homology will give a Legendrian invariant.