Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold

TL;DR

This paper provides a combinatorial method to compute Legendrian contact homology for links in certain 3-manifolds, leading to insights into symplectic homology of 4-dimensional Weinstein manifolds.

Contribution

It introduces a combinatorial description of Legendrian contact homology in specific 3-manifolds, connecting it to symplectic homology of Weinstein 4-manifolds.

Findings

Combinatorial formulas for Legendrian contact homology in $S^1\times S^2$ and connected sums.

Relation between Legendrian homology and symplectic homology via surgery formulas.

Analysis of invariance of Legendrian homology under deformations.

Abstract

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology, this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformations, from both the combinatorial and the analytical perspectives.