Augmentations and rulings of Legendrian links in $\#^k(S^1\times S^2)$

Caitlin Leverson

Vol. 288 (2017), No. 2, 381–423

DOI: 10.2140/pjm.2017.288.381

Abstract

Given a Legendrian link in ${\#}^k\left(S^1\times S^2\right)$, 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 $\mathbb{Z}\left[t,t^{-1}\right]$ 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

Milestones

Received: 10 November 2015

Revised: 2 September 2016

Accepted: 30 October 2016

Published: 28 April 2017