A bordered Legendrian contact algebra

Abstract

In A bordered Chekanov–Eliashberg algebra, Sivek proves a "van Kampen" decomposition theorem for the combinatorial Legendrian contact algebra (also known as the Chekanov-Eliashberg algebra) of knots in standard contact $\mathbb{R}^3$. We prove an analogous result for the holomorphic curve version of the Legendrian contact algebra of certain Legendrians submanifolds in standard contact $J^1(M)$. This includes all one- and two-dimensional Legendrians, and some higher-dimensional ones. We present various applications including a Mayer-Vietoris sequence for linearized contact homology similar to A bordered Chekanov–Eliashberg algebra and a connect sum formula for the augmentation variety introduced in L. Ng, Framed knot contact homology. The main tool is the theory of gradient flow trees developed in T. Ekholm, Morse flow trees and Legendrian contact homology in 1-jet spaces.