Distinguishing Legendrian knots in topological types 74, 948, 10136 with maximal Thurston–Bennequin number

Abstract

In a recent work of Dynnikov and Prasolov [I. Dynnikov and M. Prasolov, An algorithm for comparing Legendrian knots, preprint (2023), arXiv:2309.05087] a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method, we confirm conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997] about Legendrian knots in topological types [Formula: see text], [Formula: see text] and [Formula: see text]. This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.