Rectangular diagrams of links, surfaces, and foliations

The formalism of rectangular diagrams provides a convenient framework for representing links and surfaces in the three-sphere in a consistent manner. They appear to have a strong connection with contact topology, as well as nice combinatorial properties, which make them useful for solving algorithmic and classification problems of knot theory. In particular, they allow to construct a simple algorithm for recognizing the unknot and to solve the problem of algorithmic classification of Legendrian and transverse links. One can also use rectangular diagrams to obtain explicit presentations of finite depth taut foliations in link complements.