In recent joint works with Maxim Prasolov and Vladimir Shastin we developed a method to decide algorithmically whether or not two given Legendrian knots are equivalent. The method is based on the formalism of rectangular diagrams, which we have extended to Giroux' convex surfaces and Legendrian graphs. In many cases, the approach allows to distinguish Legendrian knots that are not distinguished by other known means or have so large complexity that other methods are not feasible to apply to them. We provide an example of two non-equivalent Legendrian knots that cobound an annulus embedded in the three-sphere and tangent to the standard contact structure along the entire boundary. Such examples have not been known earlier.