Abstract:  Legendrian knots are particular closed curves in 3-dimensional contact manifolds that play an important role in contact and symplectic topology.   In this talk, I will focus on Legendrian knots in R^3 with its standard contact structure.  The study of such knots provides an interesting variation on the classical theory of knotted circles in 3-dimensional space.

After discussing some basic aspects of Legendrian knot theory, a type of decomposition of the front diagram of a Legendrian knot called a normal ruling will be introduced.  This notion is due independently to Chekanov and Pushkar, and Fuchs.  Along with being visually appealing, normal rulings turn out to be connected with other interesting constructions from symplectic topology and low-dimensional topology.  In particular, I will discuss results relating normal rulings with generating families, knot polynomials, and Legendrian contact homology.