  In this paper, we present several new structures for the colored HOMFLY-PT
invariants of framed links. First, we prove the strong integrality property for
the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein
theory developed by H. Morton and his collaborators. By this strong integrality
property, we immediately obtain several symmetric properties for the full
colored HOMFLY-PT invariants of links. Then, we apply our results to refine the
mathematical structures appearing in the Labastida-Mari\~no-Ooguri-Vafa (LMOV)
integrality conjecture for framed links. As another application of the strong
integrality, we obtain that the $q=1$ and $a=1$ specializations of the
normalized colored HOMFLY-PT invariant are well-defined link polynomials. We
find that a conjectural formula for the colored Alexander polynomial which is
the $a=1$ specialization of the normalized colored HOMFLY-PT invariant implies
that a special case of the LMOV conjecture for frame knot holds.
