Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

We define an algebraic/combinatorial object on the front projection $\mathrm{\Sigma }$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on $\mathrm{\Sigma }$ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of $L_{\mathrm{\Sigma }}$, where $L_{\mathrm{\Sigma }}$ is the Ng resolution of $\mathrm{\Sigma }$. In the case of Legendrian knot classes admitting representatives with two-bridge front projections, this map is bijective. We also exhibit two standard forms for MCSs and give explicit algorithms for finding these forms.