  Let $\Lambda$ be a link of Legendrian spheres in the boundary of a
subcritical $2n$-dimensional Weinstein manifold $X$. We show that, under some
geometrical assumptions, the computation of the Legendrian contact homology of
$\Lambda$ can be reduced to a computation of Legendrian contact homology in
1--jet spaces. Since the Legendrian contact homology in 1--jet spaces is well
studied, this gives a simplified way to compute the Legendrian contact homology
of $\Lambda$. We restrict to the case when the attaching spheres of the
subcritical handles of $X$ do not interact with each other, and we assume that
there are no handles of index $n-1$. Moreover, we will only consider mod 2
coefficients for now. The more general situation will be addressed in a
forthcoming paper. As an application we compute the homology of the free loop
space of $\mathbb{CP}^2$.
