Diagrammatic approach to the classification of knots and contact topology

About two decades ago I discovered that the problem of recognizing the unknot could be solved using a monotonic simplification procedure for certain type knot diagrams, which I called rectangular. They are also now known as grid diagrams. For arbitrary knots, the procedure does not work as well as it does for the unknot, since there may be many non-equivalent rectangular diagrams representing the same knot type that are not subject to any simplification by elementary moves. However, for many knot types, the set of non-simplifiable diagrams is finite, and it does not seem impossible that this is true for all knots. The classification of non-simplifiable rectangular diagrams turns out to be deeply related to the classification of Legendrian knots. This relation has been recently studied in our joint works with Maxim Prasolov and Vladimir Shastin. As a by-product of this study, we obtained an algorithm for comparing Legendrian knots, which has not been known before. In my talk, I will overview the most recent results on the subject.