A filtered generalization of the Chekanov-Eliashberg algebra

Abstract

We define a new algebra associated to a Legendrian submanifold $Λ$ of a contact manifold of the form $\mathbb{R}_{t} \times W$, called the planar diagram algebra and denoted $PDA(Λ, \mathcal{P})$. It is a non-commutative, filtered, differential graded algebra whose filtered stable tame isomorphism class is an invariant of $Λ$ together with a partition $\mathcal{P}$ of its connected components. When $Λ$ 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.