Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

TL;DR

This paper introduces Morse complex sequences (MCS), a new combinatorial tool linking generating families, rulings, and augmentations of Legendrian knots, with algorithms for standard forms and a bijection in specific cases.

Contribution

It defines MCS as a novel combinatorial object connecting various invariants of Legendrian knots and establishes a surjective map to augmentation classes, with a bijection for two-bridge front projections.

Findings

Defined Morse complex sequences (MCS) for Legendrian knots.

Established a surjective map from MCS classes to augmentation classes.

Provided algorithms for standard forms of MCSs and proved bijection in two-bridge cases.

Abstract

We define an algebraic/combinatorial object on the front projection $\Sigma$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on $\Sigma$ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of $L_\Sigma$, where $L_\Sigma$ is the Ng resolution of $\Sigma$. In the case of Legendrian knot classes admitting representatives with two-bridge front projections, this map is bijective. We also exhibit two standard forms for MCSs and give explicit algorithms for finding these forms.