  Sinha and Vafa \cite {sinha} 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 \cite{mar9} based on the work of Morton and Ryder\cite{mor} 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 $c=0$ and $c=2$ giving the
open-string topological amplitude on the orientifold of the resolved conifold.
