A combinatorial DGA for Legendrian knots from generating families
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.
Keywords
Cite
@article{arxiv.1106.3357,
title = {A combinatorial DGA for Legendrian knots from generating families},
author = {Michael B. Henry and Dan Rutherford},
journal= {arXiv preprint arXiv:1106.3357},
year = {2015}
}
Comments
60 pages, 46 figures; v2. Exposition improved, minor error in the proof of Thm 5.5 corrected; to appear in Commun. Contemp. Math