The authors introduce what they call a full colored HOMFLYPT invariant of a link with each component colored by a pair of partitions, it is essentially the usual framed invariant applied to the satellite link, and prove a number of results for its variants. For example, for a normalized version at the limit \(q\to1\), which is called the composite invariant, it is proved that the growth in \(a\) is exponential with the sizes of coloring partitions in the exponent.
But most of the paper is dedicated to versions of the framed Labastida-Mariño-Ooguri-Vafa (LMOV) integrality conjecture. For the composite invariant the conjecture is checked for \(T(2,2k)\) torus links with small framing. Then a reformulated framed composite invariant \(\check{\mathcal{R}}_{\vec{\mu}}\) is introduced (it is related to another one from the authors’ previous work with K. Liu and P. Peng [“Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials”, Preprint, arXiv:1402.3571]), and it is proved that it belongs to \(\mathbb{Z}[z^2,a^{\pm1}]\). For the case \(\vec{\mu}=((p)\dots(p))\) with \(p\) prime the authors propose a conjectural congruence skein relation for \(\check{\mathcal{R}}_{\vec{\mu}}\) that happens to be identical to the one they previously derived for a reformulated colored Kauffman invariant. This conjecture is tested on a number of examples, including the unknot with \(2k+1\) negative kinks, and \(T(2,2k)\), \(T(2,2k+1)\) and \(T(2,2k+2)\) torus links.