• geometria di contatto
• invarianti classici
• nodi Legendriani
• struttura di contatto Tesa

This thesis is about topologically trivial Legendrian knots in an arbitrary 3-manifold, equipped with a contact structure that is tight and positively cooriented. We classify them up to contact isotopy, through the theory of characteristic foliations and planar graphs. The following work is based on a Eliashberg and Fraser’s paper of 1998, republished in 2009.

In the first chapter we present the basic notions and results. We fix the concept of isotropic submanifold, and that of Legendrian knot. Moreover, we define the classical invariants used for the classification: the Thurston-Bennequin and Maslov numbers. We conclude by stating the classification theorem, and by presenting the catalogue of Legendrian knots used as prototype.

In the second chapter we discuss the concept of singular foliation. To each Legendrian knot bounding a disk we assign a specific foliation, which is iteratively reduced to obtain a standard form called “exceptional elliptic form”. This operation is carried thought conversion lemmas, which had not been fully published until 2009.

In the last chapter we assign to every topologically trivial Legendrian knot a planar tree, called skeleton, and we connect the theory of such objects to our setting. Finally, by proving that a contact isotopy of such skeletons implies an analogous one between the associated Legendrian knots, we can conclude the proof of the classification theorem.