Legendrian Grid Number One Knots and Augmentations of Their Differential Algebras

In this article we study the differential graded algebra (DGA) invariant associated to Legendrian knots in tight lens spaces. Given a grid number one diagram for a knot in L(p,q), we show how to construct a special Lagrangian diagram suitable for computing the DGA invariant for the Legendrian knot specified by the diagram. We then specialize to L(p,p−1) and show that for two families of knots, the existence of an augmentation of the DGA depends solely on the value of p.