  Sinha and Vafa had conjectured that the $SO$ Chern-Simons gauge theory on
$S^3$ must be dual to the closed $A$-model topological string on the
orientifold of a resolved conifold. Though the Chern-Simons free energy could
be rewritten in terms of the topological string amplitudes providing evidence
for the conjecture, we needed a novel idea in the context of Wilson loop
observables to extract cross-cap $c=0,1,2$ topological amplitudes. Recent paper
of Marino based on the work of Morton and Ryder has clearly shown that the
composite representation placed on the knots and links plays a crucial role to
rewrite the topological string cross-cap $c=0$ amplitude. This enables
extracting the unoriented cross-cap $c=2$ topological amplitude. In this paper,
we have explicitly worked out the composite invariants for some framed knots
and links carrying composite representations in U(N) Chern-Simons theory. We
have verified generalised Rudolph's theorem, which relates composite invariants
to the invariants in SO(N) Chern-Simons theory, and also verified Marino's
conjectures on the integrality properties of the topological string amplitudes.
For some framed knots and links, we have tabulated the BPS integer invariants
for cross-cap $c=0$, $c=1$ and $c=2$ giving the open-string topological
amplitude on the orientifold of the resolved conifold.
