Let L be a Legendrian knot in R^3 with the standard contact structure. In
[10], 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.Comment: 28 pages, 17 figure
