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

Abstract

We define an algebraic/combinatorial object on the front projection $\Sigma$ 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 $\Sigma$ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of $L_{\Sigma }$, where $L_{\Sigma }$ is the Ng resolution of $\Sigma$. 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.

54 pages, 32 figures; v3. Definition of main object generalized slightly, other minor revisions, to appear in Pacific J. Math