A method for distinguishing Legendrian and transverse links

Legendrian (respectively, transverse) links are smooth links in the three-space that are tangent (respectively, transverse) to the standard contact structure. Deciding whether two such links are equivalent modulo a contactomorphism is a hard problem in general. Many topological invariants of Legendrian and transverse links are known, but they do not suffice for a classification even in the case of knots of crossing number six.
In recent joint works with Maxim Prasolov and Vladimir Shastin we developed a rectangular diagram machinery for surfaces and links in the three-space. This machinery has a tight connection with contact topology, namely with Legendrian links and Giroux's convex surfaces. We are mainly interested in studying rectangular diagrams of links that cannot be monotonically simplified by means of elementary moves. It turns out that this question is nearly equivalent to classification of Legendrian links.
The main outcome we have so far is an algorithm for comparing two Legendrian (or transverse) links. The computational complexity of the algorithm is, of course, very high, but, in many cases, certain parts of the procedure can be bypassed, which allows us to distinguish quite complicated Legendrian knots. In particular, we have managed to provide an example of two inequivalent Legendrian knots cobounding an annulus tangent to the standard contact structure along the entire boundary. Such examples were previously unknown.
The work is supported by the Russian Science Foundation under grant 19-11-00151