Rectangular diagrams and Giroux's convex surfaces

Rectangular diagrams can be considered as a special class of plane diagrams of links. Every link can be represented by a rectangular diagram and an analogue of Reidemeister theorem holds that any two rectangular diagrams of the same link are related by a sequence of elementary moves.
There is a natural complexity function on the set of rectangular diagrams for which a trivial knot can be recognized by a monotonic simplification, as shown by I. Dynnikov. Or equivalently, any rectangular diagram of a trivial knot can be transformed into a minimal one by elementary moves which do not increase the complexity.
It is convenient to represent by rectangular diagrams Legendrian links, i.e. which are tangent to the plane distribution $\ker(dz+xdy)$ in $\mathbb{R}^3.$ In a recent joint paper with I. Dynnikov it is shown that an extension of monotonic simplification to arbitrary links is closely related to a classification of Legendrian representatives in a fixed topological type.
One of the key instruments of low-dimensional contact topology, and particularly of Legendrian knot theory, is the Giroux's notion of convex surface. In our joint work with I. Dynnikov which is in preparation we show that convex surfaces in $\mathbb{R}^3$ can be nicely described in 'rectangular language'. We give an example of two Legendrian knots which can be distinguished using an analogue of rectangular diagrams for surfaces and which can not be distinguished by known algebraic invariants due to the lack of computational power.