Weak Relative Calabi-Yau Structures for Legendrian Contact Homology

Legendrian Contact Homology (LCH) and its augmentations are important invariants of Legendrian submanifolds, and for Legendrian knots in the standard contact 3-space in particular. We increase understanding of the algebraic structure of LCH by generalizing the duality isomorphism and long exact sequence for linearized LCH for Legendrian knots to a weak relative Calabi-Yau structure for $A_{\mathrm{\infty }}$ bimodules over the positive augmentation category.