Equivalence classes of augmentations and Morse complex sequences of Legendrian knots

TL;DR

This paper establishes a bijective correspondence between Morse complex sequences and augmentations of Legendrian contact homology, leading to new invariants and classifications for Legendrian knots.

Contribution

It proves the bijection between Morse complex sequences and augmentations, enhancing understanding of Legendrian knot invariants.

Findings

Homotopic augmentations determine the same graded normal ruling.

Count of Morse complex sequences is a Legendrian isotopy invariant.

Homotopic augmentations have isomorphic linearized contact homology groups.

Abstract

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.