Computing homology invariants of Legendrian knots

Abstract

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 paper 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 (MCS). First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| ≤ 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 [The correspondence between augmentations and rulings for Legendrian knots, Pacific J. Math.224(1) (2006) 141–150] need not have isomorphic homology groups.