Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials

Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulated) colored HOMFLY-PT invariants actually lie in the ring $\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Secondly, we establish some symmetric formulas for colored HOMFLY-PT invariants of links, which include the rank-level duality as an easy consequence. Finally, motivated by the Labastida-Mari\~no-Ooguri-Vafa conjecture for framed links, we propose congruent skein relations for (reformulated) colored HOMFLY-PT invariants which are the generalizations of the skein relation for classical HOMFLY-PT polynomials. Then we study the congruent skein relation for colored Jones polynomials. In fact, we obtain a succinct formula for the case of knot. As an application, we prove a vanishing result for Reshetikhin-Turaev invariants of a family of 3-manifolds. Finally we study the congruent skein relations for $SU(n)$ quantum invariants.