  In this paper, we prove that the Chekanov-Eliashberg algebra of an
horizontally displaceable n-dimensional Legendrian sphere in the contactisation
of a Liouville manifold is a (n+1)-Calabi-Yau differential graded algebra. In
particular it means that there is a quasi-isomorphism of DG-bimodules between
the diagonal bimodule and the inverse dualizing bimodule associated to the
Chekanov-Eliashberg algebra. On some cyclic version of these bimodules, which
are chain complexes computing the Hochschild homology and cohomology of the
Chekanov-Eliashberg algebra, we construct $A_\infty$ operations and show that
the Calabi-Yau isomorphism extends to a family of maps satisfying the
$A_\infty$-functor equations.
