Rectangular diagrams of surfaces: the basic moves

In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere S3 by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to prove a Reidemeister type theorem for rectangular diagrams of surfaces.

Classification of Legendrian knots of topological type 76 with maximal Thurston–Bennequin number

We classify Legendrian knots of topological type [Formula: see text] having maximal Thurston–Bennequin number confirming the corresponding conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997].

Tight contact structures via admissible transverse surgery

We investigate the line between tight and overtwisted for surgeries on fibered transverse knots in contact 3-manifolds. When the contact structure [Formula: see text] is supported by the fibered knot [Formula: see text], we obtain a characterization of when negative surgeries result in a contact structure with nonvanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure [Formula: see text] supported by the mirror knot [Formula: see text]. We derive several corollaries about the existence of tight contact structures, L-space knots outside [Formula: see text], nonplanar contact structures, and nonplanar Legendrian knots.