Full colored HOMFLYPT invariants, composite invariants and congruence skein relations

In this paper, we investigate the properties of certain quantum invariants of links by using the HOMFLY skein theory. First, we obtain the limit behavior for the full colored HOMFLYPT invariant which is the natural generalization of the colored HOMFLYPT invariant. Then we focus on the composite invariant which is a certain combination of the full colored HOMFLYPT invariants. Motivated by the study of the Labastida–Mariño–Ooguri–Vafa conjecture for the framed composite invariants of links, we introduce the notion of reformulated composite invariant Rˇ _p(L; q, a). By using the HOMFLY skein theory, we prove that Rˇ _p(L; q, a) actually lies in the integral ring 2Z[(q-q-1)2,a±1]. Finally, we propose a conjectural congruence skein relation for Rˇ _p(L; q, a) and prove it for certain special cases.