  We started a program to study the open string integrality invariants (LMOV
invariants) for toric Calabi-Yau 3-folds with Aganagic-Vafa brane (AV-brane)
several years ago. This paper is devoted to the case of resolved conifold with
one out AV-brane in any integer framing $\tau$, which is the large $N$ duality
of Chern-Simons theory for a framed unknot with integer framing $\tau$ in
$S^3$. By using the methods from string dualities, we compute several explicit
formulae of the corresponding LMOV invariants for this special model, whose
integrality properties have been proved in a separated paper.
