Distinguishing Legendrian knots of topological type $7_4$, $9_{48}$ and ${10}_{136}$

In a recent work of I. Dynnikov and M. Prasolov a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method we confirm conjectures of Ng and Chongchitmate about Legendrian knots in topological types $7_4$, $9_{48}$ and ${10}_{136}$. This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.