  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.
