Augmentations and Rulings of Legendrian Links in ${\#}^k(S^1\times S^2)$

Abstract

Given a Legendrian link in ${\#}^k(S^1\times S^2)$, we extend the definition of a normal ruling from $J^1(S^1)$ given by Lavrov and Rutherford and show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over $\mathbb{Z}[t,t^{-1}]$ is equivalent to the existence of a normal ruling of the front diagram. For Legendrian knots, we also show that any even graded augmentation must send $t$ to $-1$. We use the correspondence to give nonvanishing results for the symplectic homology of certain Weinstein $4$-manifolds. We show a similar correspondence for the related case of Legendrian links in $J^1(S^1)$, the solid torus.

30 pages, 16 figures

Cited by in corpus (3)

• Legendrian Fronts for Affine Varieties
• Simplifying Weinstein Morse functions
• Exact Lagrangian tori in symplectic Milnor fibers constructed with fillings