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 ${\mathbb{R}}_t\times \mathrm{W}$, called the planar diagram algebra, and denote it by $\mathrm{PDA}(\Lambda ,\mathcal{P})$. 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, $\mathrm{PDA}$ is the Chekanov–Eliashberg algebra. In general, the $\mathrm{PDA}$ differential counts holomorphic disks with multiple positive punctures using a combinatorial framework inspired by string topology.