A filtered generalization of the Chekanov-Eliashberg algebra

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.

59 pages, 19 figures. V5 - minor corrections