A filtered generalization of the Chekanov-Eliashberg algebra

TL;DR

This paper introduces the planar diagram algebra, a new invariant for Legendrian submanifolds that generalizes the Chekanov-Eliashberg algebra by incorporating multiple positive punctures through a combinatorial approach.

Contribution

It defines the PDA, a filtered differential graded algebra that extends the Chekanov-Eliashberg algebra to disconnected Legendrians with a combinatorial disk-counting method.

Findings

PDA is an invariant of Legendrian submanifolds with a partition.

PDA generalizes the Chekanov-Eliashberg algebra for disconnected Legendrians.

The differential counts holomorphic disks with multiple positive punctures.

Abstract

We define a new algebra associated to a Legendrian submanifold $\Lambda$ of a contact manifold of the form $\mathbb{R}_{t} \times W$, called the planar diagram algebra and denoted $PDA(\Lambda, \mathcal{P})$. It is a non-commutative, filtered, differential graded algebra whose filtered stable tame isomorphism class is an invariant of $\Lambda$ together with a partition $\mathcal{P}$ of its connected components. When $\Lambda$ is connected, $PDA$ is the Chekanov-Eliashberg algebra. In general, the $PDA$ differential counts holomorphic disks with multiple positive punctures using a combinatorial framework inspired by string topology.