Due to Alexander, it is well known that every closed
oriented 3-manifold has an open book decomposition. In this talk, we
will discuss the importance of the open book decompositions in
manifold theory, in paticular in contact geometry. After a brief
introduction on contact 3-manifolds, we will focus on a class of knots
called Legendrian knots. We will define a new invariant for Legendrian
knots using open book decompositions. We define the support genus
sg(L) of a Legendrian knot L in a contact manifold M as the minimal
genus of a page of an open book of M supporting the contact structure
such that L sits on a page and the framings given by the contact
structure and the page agree. We will discuss the applications of this
invariant and list several open problems related to support genus of
Legendrian knots.