Alberto Cavallo:
Adapted open book decompositions and Legendrian Heegaard diagrams for links

Abstract:
We describe a condition for an open book decomposition of a 3-manifold M,
supporting the contact structure xi, to be compatible with a given Legendrian
link L in (M, xi).  This is done by taking a ribbon S of the 1-skeleton G of a
contact cell decomposition of (M, xi), where G contains L, and applying a
result of Giroux together with an appropriate collection of arcs in S.
Furthermore, we show that we can construct a special type of Heegaard
diagrams, that we call Legendrian Heegaard diagrams, from such adapted open
books.  At the end, we briefly describe how to define an invariant using the
representation of Legendrian links that we introduced before; generalizing a
result on Legendrian knots of Lisca, Ozsváth, Stipsicz and Szabó.