We define an algebraic/combinatorial
object on the front projection Σ 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 Σ and construct a surjective map from the
equivalence classes to the set of chain homotopy classes of augmentations of LΣ,
where LΣ is the Ng resolution of Σ. 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. The definition of an MCS, the equivalence relation, and the
statements of some of the results originate from unpublished work of Petya
Pushkar.
Keywords
Legendrian knots, augmentations, normal rulings, generating
families, contact homology
