New structures for colored HOMFLY-PT invariants

Abstract: 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.