Legendrian knots in contact 3-manifolds form a richer family than classical knots, due to the fact that there exist several distinct Legendrian realizations of the same topological knot type. Despite significant progress over the last few decades, the complete classification of Legendrian knots remains a distant goal. A central question is how many distinct Legendrian realizations exist for a given knot type with prescribed classical invariants. If there is only one, we call the knot type Legendrian simple. Since the early 2000s, it has been known that Legendrian non-simple knot types exist. In 2019, using knot Floer homology and the contact invariant, we established lower bounds on the number of distinct realizations, identifying new examples of Legendrian non-simple knot types. In this talk, I will present an approach for establishing upper bounds as well. I will introduce the necessary concepts from convex surface theory, a powerful tool in contact topology, and explain how we can use these techniques to classify Legendrian knots with respect to Legendrian simplicity. As an application, I will present a joint result with Vera Vértesi, which provides an upper bound on the number of distinct Legendrian realizations of certain double twist knots with maximal Thurston-Bennequin invariant.