#### Vol. 288, No. 2, 2017

 Recent Issues Vol. 289: 1 Vol. 288: 1  2 Vol. 287: 1  2 Vol. 286: 1  2 Vol. 285: 1  2 Vol. 284: 1  2 Vol. 283: 1  2 Vol. 282: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Officers Special Issues Submission Guidelines Submission Form Subscriptions Contacts Author Index To Appear ISSN: 0030-8730
Augmentations and rulings of Legendrian links in $\#^k(S^1\times S^2)$

### Caitlin Leverson

Vol. 288 (2017), No. 2, 381–423
##### Abstract

Given a Legendrian link in ${#}^{k}\phantom{\rule{0.3em}{0ex}}\left({S}^{1}×{S}^{2}\right)$, we extend the definition of a normal ruling from ${J}^{1}\left({S}^{1}\right)$ given by Lavrov and Rutherford and show that the existence of an augmentation to any field of the Chekanov–Eliashberg differential graded algebra over $ℤ\left[t,{t}^{-1}\right]$ 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 ${J}^{1}\left({S}^{1}\right)$, the solid torus.

##### Keywords
Legendrian knot, Legendrian submanifold, contact manifold, normal ruling
##### Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 53D42, 57M27