Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

Abstract

Let $L$ be a Legendrian knot in ${ℝ}^3$ with the standard contact structure. In earlier work of Henry, a map was constructed from equivalence classes of Morse complex sequences for $L$, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of $L$. Moreover, this map was shown to be a surjection. We show that this correspondence is, in fact, a bijection. As a corollary, homotopic augmentations determine the same graded normal ruling of $L$ and have isomorphic linearized contact homology groups. A second corollary states that the count of equivalence classes of Morse complex sequences of a Legendrian knot is a Legendrian isotopy invariant.