A bordered Legendrian contact algebra

TL;DR

This paper extends the decomposition theorem for Legendrian contact algebras to a holomorphic curve setting, covering various Legendrian submanifolds and providing new computational tools and formulas.

Contribution

It proves a van Kampen type decomposition for the holomorphic curve version of Legendrian contact algebra, broadening the scope beyond knots in ^3 and including higher-dimensional cases.

Findings

Establishes a Mayer-Vietoris sequence for linearized contact homology.

Derives a connect sum formula for the augmentation variety.

Applies gradient flow trees to prove the main results.

Abstract

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 $\R^3$ . 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 1- and 2-dimensional Legendrians, and some higher dimensional ones. We present various applications including a Mayer-Vietoris sequence for linearized contact homology similar to Sivek's and a connect sum formula for the augmentation variety introduced by Ng. The main tool is the theory of gradient flow trees developed by Ekholm.