A combinatorial DGA for Legendrian knots from generating families

TL;DR

This paper introduces a new combinatorial differential graded algebra (DGA) for Legendrian knots derived from Morse complex sequences and generating families, providing a geometric interpretation and invariance properties.

Contribution

It constructs a novel DGA based on Morse-theoretic data, linking it to existing invariants and offering a geometric perspective through gradient staircases.

Findings

The DGA counts chord paths in the front projection of Legendrian knots.

Linearized complexes are invariant under MCS equivalence.

The DGA coincides with the Chekanov-Eliashberg DGA under certain conditions.

Abstract

For a Legendrian knot L in R^3 with a chosen Morse complex sequence (MCS) we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov-Eliashberg DGA after changing coordinates by an augmentation.