An L-infinity structure for Legendrian contact homology

For any Legendrian knot or link in ${\mathbb{R}}^3$, we construct an $L_{\mathrm{\infty }}$ algebra that can be viewed as an extension of the Chekanov-Eliashberg differential graded algebra. The $L_{\mathrm{\infty }}$ structure incorporates information from rational Symplectic Field Theory and can be formulated combinatorially. One consequence is the construction of a Poisson bracket on commutative Legendrian contact homology, and we show that the resulting Poisson algebra is an invariant of Legendrian links under isotopy.