  The Chekanov-Eliashberg differential graded algebra of a Legendrian knot L is
a rich source of Legendrian knot invariants, as is the theory of generating
families. The set P(L) of homology groups of augmentations of the
Chekanov-Eliashberg algebra is an invariant, as is a count of objects from the
theory of generating families called graded normal rulings. This article gives
two results demonstrating the usefulness of computing the homology group of an
augmentation using a combinatorial interpretation of a generating family called
a Morse complex sequence. First, we show that if the projection of L to the
xz-plane has exactly 4 cusps, then |P(L)| is less than or equal to 1. Second,
we show that two augmentations associated to the same graded normal ruling by
the many-to-one map between augmentations and graded normal rulings defined by
Ng and Sabloff need not have isomorphic homology groups.
